This document presents an algorithm for calculating the number of spanning trees in chained graphs. It begins by reviewing relevant graph theory concepts like planar graphs, spanning trees, and recursive formulas for counting spanning trees using deletion/contraction and splitting methods. It then derives explicit recursions for counting spanning trees in families of graphs like wheel graphs, fan graphs, and corn graphs. The main result is a theorem providing a system of equations to calculate the number of spanning trees in a chained graph based on splitting it into components and accounting for the connecting paths. Applications to counting spanning trees in chained wheel graphs and chained corn graphs are discussed.
THE RESULT FOR THE GRUNDY NUMBER ON P4- CLASSESgraphhoc
This document summarizes research on calculating the Grundy number of fat-extended P4-laden graphs. It begins by introducing the Grundy number and discussing that it is NP-complete to calculate for general graphs. It then presents previous work that has found polynomial time algorithms to calculate the Grundy number for certain graph classes. The main results are that the document proves that the Grundy number can be calculated in polynomial time, specifically O(n3) time, for fat-extended P4-laden graphs by traversing their modular decomposition tree. This implies the Grundy number can also be calculated efficiently for several related graph classes that are contained within fat-extended P4-laden graphs.
Graph Dynamical System on Graph ColouringClyde Shen
This document discusses using a graph dynamical system to model the vertex coloring problem. It proposes using a Graph-cellular Automaton (GA), which is an extension of cellular automata, to distribute the coloring process. In the GA model, each vertex independently chooses its optimal color based on the colors of its adjacent vertices. The document outlines testing the GA approach on different types of graphs and analyzing whether it can find colorings for them. It provides background on graph coloring, planar graphs, and introduces the graph dynamical system and GA models.
The document discusses finite fields and related algebraic concepts. It begins by defining groups, rings, and fields. It then focuses on finite fields, particularly GF(p) fields consisting of integers modulo a prime p. It discusses finding multiplicative inverses in such fields using the extended Euclidean algorithm. As an example, it finds the inverse of 550 modulo 1759.
Basic galois field arithmatics required for error control codesMadhumita Tamhane
Knowledge of Galois Fields is must for understanding Error Control Codes. This presentation undertakes concepts of Galois Field required for understanding Error Control Codes in very simple manner, explaining its complex mathematical intricacies in a structured manner.
This document summarizes research on embedding planar point sets with integral squared distances in lattices corresponding to rings of integers in imaginary quadratic fields. It is shown that if the squared distance between at least one pair of points is integral, and the ring of integers has the property that the square of every ideal is principal, then the point set can be embedded in the ring of integers. Additionally, if the point set is primitive with relatively prime squared distances, and the ring of integers is a principal ideal domain, then the point set embeds in the ring of integers. Examples of embeddings in specific rings of integers are provided.
Fractional integration and fractional differentiation of the product of m ser...Alexander Decker
This document presents theorems on fractional integrals and derivatives of the product of an M-series and H-function. An M-series is a special case of an H-function, and represents important functions in physics and applied sciences. Theorems are derived for the Riemann-Liouville fractional integral and derivative of the product. Additional theorems provide formulas for fractional integrals of the M-series alone, defined using an H-function operator previously established by Saxena and Khumbhat. The theorems extend previous work and provide new formulas incorporating these important special functions.
Total Dominating Color Transversal Number of Graphs And Graph Operationsinventionjournals
Total Dominating Color Transversal Set of a graph is a Total Dominating Set of the graph which is also Transversal of Some 휒 - Partition of the graph. Here 휒 is the Chromatic number of the graph. Total Dominating Color Transversal number of a graph is the cardinality of a Total Dominating Color Transversal Set which has minimum cardinality among all such sets that the graph admits. In this paper, we consider the well known graph operations Join, Corona, Strong product and Lexicographic product of graphs and determine Total Dominating Color Transversal number of the resultant graphs.
The document is a research paper that presents new results on odd harmonious graphs. It introduces the concepts of m-shadow graphs and m-splitting graphs. The paper proves that m-shadow graphs of paths and complete bipartite graphs are odd harmonious for all m ≥ 1. It also proves that n-splitting graphs of paths, stars and symmetric products of paths and null graphs are odd harmonious for all n ≥ 1. Additional families of graphs, including m-shadow graphs of stars and various graph constructions using paths, stars and their splitting graphs, are shown to admit odd harmonious labeling.
THE RESULT FOR THE GRUNDY NUMBER ON P4- CLASSESgraphhoc
This document summarizes research on calculating the Grundy number of fat-extended P4-laden graphs. It begins by introducing the Grundy number and discussing that it is NP-complete to calculate for general graphs. It then presents previous work that has found polynomial time algorithms to calculate the Grundy number for certain graph classes. The main results are that the document proves that the Grundy number can be calculated in polynomial time, specifically O(n3) time, for fat-extended P4-laden graphs by traversing their modular decomposition tree. This implies the Grundy number can also be calculated efficiently for several related graph classes that are contained within fat-extended P4-laden graphs.
Graph Dynamical System on Graph ColouringClyde Shen
This document discusses using a graph dynamical system to model the vertex coloring problem. It proposes using a Graph-cellular Automaton (GA), which is an extension of cellular automata, to distribute the coloring process. In the GA model, each vertex independently chooses its optimal color based on the colors of its adjacent vertices. The document outlines testing the GA approach on different types of graphs and analyzing whether it can find colorings for them. It provides background on graph coloring, planar graphs, and introduces the graph dynamical system and GA models.
The document discusses finite fields and related algebraic concepts. It begins by defining groups, rings, and fields. It then focuses on finite fields, particularly GF(p) fields consisting of integers modulo a prime p. It discusses finding multiplicative inverses in such fields using the extended Euclidean algorithm. As an example, it finds the inverse of 550 modulo 1759.
Basic galois field arithmatics required for error control codesMadhumita Tamhane
Knowledge of Galois Fields is must for understanding Error Control Codes. This presentation undertakes concepts of Galois Field required for understanding Error Control Codes in very simple manner, explaining its complex mathematical intricacies in a structured manner.
This document summarizes research on embedding planar point sets with integral squared distances in lattices corresponding to rings of integers in imaginary quadratic fields. It is shown that if the squared distance between at least one pair of points is integral, and the ring of integers has the property that the square of every ideal is principal, then the point set can be embedded in the ring of integers. Additionally, if the point set is primitive with relatively prime squared distances, and the ring of integers is a principal ideal domain, then the point set embeds in the ring of integers. Examples of embeddings in specific rings of integers are provided.
Fractional integration and fractional differentiation of the product of m ser...Alexander Decker
This document presents theorems on fractional integrals and derivatives of the product of an M-series and H-function. An M-series is a special case of an H-function, and represents important functions in physics and applied sciences. Theorems are derived for the Riemann-Liouville fractional integral and derivative of the product. Additional theorems provide formulas for fractional integrals of the M-series alone, defined using an H-function operator previously established by Saxena and Khumbhat. The theorems extend previous work and provide new formulas incorporating these important special functions.
Total Dominating Color Transversal Number of Graphs And Graph Operationsinventionjournals
Total Dominating Color Transversal Set of a graph is a Total Dominating Set of the graph which is also Transversal of Some 휒 - Partition of the graph. Here 휒 is the Chromatic number of the graph. Total Dominating Color Transversal number of a graph is the cardinality of a Total Dominating Color Transversal Set which has minimum cardinality among all such sets that the graph admits. In this paper, we consider the well known graph operations Join, Corona, Strong product and Lexicographic product of graphs and determine Total Dominating Color Transversal number of the resultant graphs.
The document is a research paper that presents new results on odd harmonious graphs. It introduces the concepts of m-shadow graphs and m-splitting graphs. The paper proves that m-shadow graphs of paths and complete bipartite graphs are odd harmonious for all m ≥ 1. It also proves that n-splitting graphs of paths, stars and symmetric products of paths and null graphs are odd harmonious for all n ≥ 1. Additional families of graphs, including m-shadow graphs of stars and various graph constructions using paths, stars and their splitting graphs, are shown to admit odd harmonious labeling.
This document discusses theorems related to list coloring of graphs. It begins with definitions of graph coloring, list coloring, and other graph theory concepts. It then discusses several important theorems in the area, including:
- Thomassen's 5-list coloring theorem, which states that planar graphs are 5-list colorable.
- A result showing that the list chromatic number of a graph is bounded above by the maximum degree plus one.
- Examples showing that planar graphs require lists of size at least 4, as 4-list coloring is not always possible.
- A relationship between list coloring and sum-list coloring, and results relating their parameters.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
Dynamic Programming Over Graphs of Bounded TreewidthASPAK2014
This document discusses treewidth and algorithms for graphs with bounded treewidth. It contains three parts:
1) Algorithms for bounded treewidth graphs, including showing weighted max independent set and c-coloring can be solved in FPT time parameterized by treewidth using dynamic programming on tree decompositions.
2) Graph-theoretic properties of treewidth such as definitions of treewidth and tree decompositions.
3) Applications of these algorithms to problems like Hamiltonian cycle on general graphs.
On the equality of the grundy numbers of a graphijngnjournal
Our work becomes integrated into the general problem of the stability of the network ad hoc. Some, works attacked(affected) this problem. Among these works, we find the modelling of the network ad hoc in the form of a graph. Thus the problem of stability of the network ad hoc which corresponds to a problem of allocation of frequency amounts to a problem of allocation of colors in the vertex of graph. we present use a parameter of coloring " the number of Grundy”. The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedy k-coloring, that is a coloring with colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic, Cartesian and direct products of two graphs in terms of the Grundy numbers of these graphs.
This document discusses important cuts and separators in graphs and their algorithmic applications. It begins by defining cuts, minimum cuts, and minimal cuts. It then characterizes cuts as edges on the boundary of a set of vertices. The document discusses properties of cuts like submodularity. It introduces the concept of important cuts and separators, proving several properties about them. Importantly, it proves that the number of important cuts of size at most k is at most 4k. The document discusses applications of important cuts and separators to parameterized algorithms.
Cs6702 graph theory and applications Anna University question paper apr may 2...appasami
This document contains a past exam paper for the subject "Graph Theory and Applications". It includes 15 multiple choice and long answer questions covering topics such as Euler graphs, planar graphs, spanning trees, matchings, counting principles, and graph algorithms. The key provided contains concise definitions and explanations for the concepts and steps to solve problems in 3 sentences or less.
The document presents tables of prime numbers with interesting digit patterns that generalize repunit numbers, which are numbers composed of only the digit 1. It identifies several new prime numbers discovered through testing numbers of the forms presented. Notably, it proves that the repunit number (10^317 - 1)/9 is prime, adding to the list of known repunit primes.
Connected domination in block subdivision graphs of graphsAlexander Decker
This document discusses connected domination in block subdivision graphs. It begins by defining key terms such as domination, connected domination, block subdivision graphs, and connected domination number. It then presents three theorems that establish bounds on the connected domination number of block subdivision graphs in terms of the number of vertices and cut vertices of the original graph G.
function f is said to be an even harmonious labeling of a graph G with q edges if f is an injection from the vertices of G to the integers from 0 to 2q and the induced function f* from the edges ofG to {0, 2…….….2(q-1)} defined by f* (uv) = f (u) +f (v) (mod 2q) is bijective. The graph G is said to have an even harmonious labeling. In this paper the even harmonious labeling of a class of graph namely H (2n, 2t+1) is established.
A common fixed point of integral type contraction in generalized metric spacessAlexander Decker
This document presents a common fixed point theorem for two self-mappings S and T on a G-metric space X that satisfies a contractive condition of integral type. It begins with definitions related to G-metric spaces and contractive conditions. It then states Theorem 1.1, which proves that if S and T satisfy the given integral type contractive condition, along with other listed conditions, then S and T have a unique point of coincidence in X. If S and T are also weakly compatible, then they have a unique common fixed point. The proof of Theorem 1.1 is then provided.
The document discusses applications of graphs with bounded treewidth. It covers the following key points:
1) Courcelle's theorem shows that many NP-complete graph problems can be solved in linear time for graphs of bounded treewidth using monadic second-order logic. This includes problems like independent set, coloring, and Hamiltonian cycle.
2) The treewidth of a graph is closely related to its largest grid minor - graphs with large treewidth contain large grid minors. There are polynomial relationships between treewidth and largest grid minor for planar graphs.
3) Planar graphs have bounded treewidth if and only if they exclude some grid configuration as a contraction. This helps characterize planar graphs of bounded treewidth
Enumeration methods are very important in a variety of settings, both mathematical and applications. For many problems there is actually no real hope to do the enumeration in reasonable time since the number of solutions is so big. This talk is about how to compute at the limit.
The talk is decomposed into:
(a) Regular enumeration procedure where one uses computerized case distinction.
(b) Use of symmetry groups for isomorphism checks.
(c) The augmentation scheme that allows to enumerate object up to isomorphism without keeping the full list in memory.
(d) The homomorphism principle that allows to map a complex problem to a simpler one.
This document provides a tutorial on building concept lattices from numerical data using pattern structures. It introduces pattern structures as a way to handle numerical data in Formal Concept Analysis without needing to binarize the data. Pattern structures allow considering similarity between values using a similarity relation and meet operator on interval patterns. The document outlines key concepts of pattern structures, how numerical data can be treated as pattern structures, and how a similarity relation can be introduced to group similar objects in concepts. It also discusses ways to project patterns to change the granularity of the concept lattice.
Fixed point result in menger space with ea propertyAlexander Decker
This document presents a fixed point theorem for four self-maps in a Menger space. It begins by defining key concepts related to Menger spaces including probabilistic metric spaces, t-norms, neighborhoods, convergence, Cauchy sequences, and completeness. It then introduces properties like weakly compatible maps, property (EA), and JSR mappings. The main result, Theorem 3.1, proves the existence of a common fixed point for four self-maps under conditions that the map pairs satisfy a common property (EA) and are closed, JSR mappings satisfying an inequality involving the probabilistic distance functions.
Let G= (V, E) be a graph with p vertices and q edges. A graph G={V, E} with p vertices
and q edges is said to be a Permutation labelling graph if there exists a bijection function f
from set of all vertices
V G( )
to
{1, 2,3... }p such that the induced edge labelling function
h E G N : ( ) → is defined as
( ) ( ) 1 2 1 ( 2 )
,
f x h x x f x P =
or
( ) 2 f x( 1 )
f x P according as
f x f x ( 1 2 ) ( ) or
f x f x ( 2 1 ) ( ) where P is the permutation of objects( representing the
labels assigned to vertices). We in this paper have identified (m,n) Kite graph and attached
an edge to form a join to the kite graph and proved that the joins of (m,n) kite graphs is
permutation labelling graph and also have obtained some important results connecting the
joins of a (m,n) kite graph.
The document discusses the Fundamental Theorem of Calculus, which has two parts. Part 1 establishes the relationship between differentiation and integration, showing that the derivative of an antiderivative is the integrand. Part 2 allows evaluation of a definite integral by evaluating the antiderivative at the bounds. Examples are given of using both parts to evaluate definite integrals. The theorem unified differentiation and integration and was fundamental to the development of calculus.
This chapter discusses functions and graphs. It introduces different types of functions including constant, special, rational, and absolute value functions. It covers combining functions using addition, subtraction, multiplication, and composition. Inverse functions and their properties are explained. Graphing functions in a rectangular coordinate system and finding intercepts is demonstrated. The chapter also discusses symmetry of graphs about the x-axis, y-axis, origin, and line y = x.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document introduces the concept of weak triple connected domination number (γwtc) of a graph. A subset S of vertices is a weak triple connected dominating set if S is a weak dominating set and the induced subgraph <S> is triple connected. The γwtc is defined as the minimum cardinality of such a set. Some standard graphs are used to illustrate the concept and determine this number. Bounds on γwtc are obtained for general graphs, and its relationship to other graph parameters are investigated. The paper aims to develop this new graph invariant and establish basic results about weak triple connected domination.
This document discusses theorems related to list coloring of graphs. It begins with definitions of graph coloring, list coloring, and other graph theory concepts. It then discusses several important theorems in the area, including:
- Thomassen's 5-list coloring theorem, which states that planar graphs are 5-list colorable.
- A result showing that the list chromatic number of a graph is bounded above by the maximum degree plus one.
- Examples showing that planar graphs require lists of size at least 4, as 4-list coloring is not always possible.
- A relationship between list coloring and sum-list coloring, and results relating their parameters.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
Dynamic Programming Over Graphs of Bounded TreewidthASPAK2014
This document discusses treewidth and algorithms for graphs with bounded treewidth. It contains three parts:
1) Algorithms for bounded treewidth graphs, including showing weighted max independent set and c-coloring can be solved in FPT time parameterized by treewidth using dynamic programming on tree decompositions.
2) Graph-theoretic properties of treewidth such as definitions of treewidth and tree decompositions.
3) Applications of these algorithms to problems like Hamiltonian cycle on general graphs.
On the equality of the grundy numbers of a graphijngnjournal
Our work becomes integrated into the general problem of the stability of the network ad hoc. Some, works attacked(affected) this problem. Among these works, we find the modelling of the network ad hoc in the form of a graph. Thus the problem of stability of the network ad hoc which corresponds to a problem of allocation of frequency amounts to a problem of allocation of colors in the vertex of graph. we present use a parameter of coloring " the number of Grundy”. The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedy k-coloring, that is a coloring with colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic, Cartesian and direct products of two graphs in terms of the Grundy numbers of these graphs.
This document discusses important cuts and separators in graphs and their algorithmic applications. It begins by defining cuts, minimum cuts, and minimal cuts. It then characterizes cuts as edges on the boundary of a set of vertices. The document discusses properties of cuts like submodularity. It introduces the concept of important cuts and separators, proving several properties about them. Importantly, it proves that the number of important cuts of size at most k is at most 4k. The document discusses applications of important cuts and separators to parameterized algorithms.
Cs6702 graph theory and applications Anna University question paper apr may 2...appasami
This document contains a past exam paper for the subject "Graph Theory and Applications". It includes 15 multiple choice and long answer questions covering topics such as Euler graphs, planar graphs, spanning trees, matchings, counting principles, and graph algorithms. The key provided contains concise definitions and explanations for the concepts and steps to solve problems in 3 sentences or less.
The document presents tables of prime numbers with interesting digit patterns that generalize repunit numbers, which are numbers composed of only the digit 1. It identifies several new prime numbers discovered through testing numbers of the forms presented. Notably, it proves that the repunit number (10^317 - 1)/9 is prime, adding to the list of known repunit primes.
Connected domination in block subdivision graphs of graphsAlexander Decker
This document discusses connected domination in block subdivision graphs. It begins by defining key terms such as domination, connected domination, block subdivision graphs, and connected domination number. It then presents three theorems that establish bounds on the connected domination number of block subdivision graphs in terms of the number of vertices and cut vertices of the original graph G.
function f is said to be an even harmonious labeling of a graph G with q edges if f is an injection from the vertices of G to the integers from 0 to 2q and the induced function f* from the edges ofG to {0, 2…….….2(q-1)} defined by f* (uv) = f (u) +f (v) (mod 2q) is bijective. The graph G is said to have an even harmonious labeling. In this paper the even harmonious labeling of a class of graph namely H (2n, 2t+1) is established.
A common fixed point of integral type contraction in generalized metric spacessAlexander Decker
This document presents a common fixed point theorem for two self-mappings S and T on a G-metric space X that satisfies a contractive condition of integral type. It begins with definitions related to G-metric spaces and contractive conditions. It then states Theorem 1.1, which proves that if S and T satisfy the given integral type contractive condition, along with other listed conditions, then S and T have a unique point of coincidence in X. If S and T are also weakly compatible, then they have a unique common fixed point. The proof of Theorem 1.1 is then provided.
The document discusses applications of graphs with bounded treewidth. It covers the following key points:
1) Courcelle's theorem shows that many NP-complete graph problems can be solved in linear time for graphs of bounded treewidth using monadic second-order logic. This includes problems like independent set, coloring, and Hamiltonian cycle.
2) The treewidth of a graph is closely related to its largest grid minor - graphs with large treewidth contain large grid minors. There are polynomial relationships between treewidth and largest grid minor for planar graphs.
3) Planar graphs have bounded treewidth if and only if they exclude some grid configuration as a contraction. This helps characterize planar graphs of bounded treewidth
Enumeration methods are very important in a variety of settings, both mathematical and applications. For many problems there is actually no real hope to do the enumeration in reasonable time since the number of solutions is so big. This talk is about how to compute at the limit.
The talk is decomposed into:
(a) Regular enumeration procedure where one uses computerized case distinction.
(b) Use of symmetry groups for isomorphism checks.
(c) The augmentation scheme that allows to enumerate object up to isomorphism without keeping the full list in memory.
(d) The homomorphism principle that allows to map a complex problem to a simpler one.
This document provides a tutorial on building concept lattices from numerical data using pattern structures. It introduces pattern structures as a way to handle numerical data in Formal Concept Analysis without needing to binarize the data. Pattern structures allow considering similarity between values using a similarity relation and meet operator on interval patterns. The document outlines key concepts of pattern structures, how numerical data can be treated as pattern structures, and how a similarity relation can be introduced to group similar objects in concepts. It also discusses ways to project patterns to change the granularity of the concept lattice.
Fixed point result in menger space with ea propertyAlexander Decker
This document presents a fixed point theorem for four self-maps in a Menger space. It begins by defining key concepts related to Menger spaces including probabilistic metric spaces, t-norms, neighborhoods, convergence, Cauchy sequences, and completeness. It then introduces properties like weakly compatible maps, property (EA), and JSR mappings. The main result, Theorem 3.1, proves the existence of a common fixed point for four self-maps under conditions that the map pairs satisfy a common property (EA) and are closed, JSR mappings satisfying an inequality involving the probabilistic distance functions.
Let G= (V, E) be a graph with p vertices and q edges. A graph G={V, E} with p vertices
and q edges is said to be a Permutation labelling graph if there exists a bijection function f
from set of all vertices
V G( )
to
{1, 2,3... }p such that the induced edge labelling function
h E G N : ( ) → is defined as
( ) ( ) 1 2 1 ( 2 )
,
f x h x x f x P =
or
( ) 2 f x( 1 )
f x P according as
f x f x ( 1 2 ) ( ) or
f x f x ( 2 1 ) ( ) where P is the permutation of objects( representing the
labels assigned to vertices). We in this paper have identified (m,n) Kite graph and attached
an edge to form a join to the kite graph and proved that the joins of (m,n) kite graphs is
permutation labelling graph and also have obtained some important results connecting the
joins of a (m,n) kite graph.
The document discusses the Fundamental Theorem of Calculus, which has two parts. Part 1 establishes the relationship between differentiation and integration, showing that the derivative of an antiderivative is the integrand. Part 2 allows evaluation of a definite integral by evaluating the antiderivative at the bounds. Examples are given of using both parts to evaluate definite integrals. The theorem unified differentiation and integration and was fundamental to the development of calculus.
This chapter discusses functions and graphs. It introduces different types of functions including constant, special, rational, and absolute value functions. It covers combining functions using addition, subtraction, multiplication, and composition. Inverse functions and their properties are explained. Graphing functions in a rectangular coordinate system and finding intercepts is demonstrated. The chapter also discusses symmetry of graphs about the x-axis, y-axis, origin, and line y = x.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document introduces the concept of weak triple connected domination number (γwtc) of a graph. A subset S of vertices is a weak triple connected dominating set if S is a weak dominating set and the induced subgraph <S> is triple connected. The γwtc is defined as the minimum cardinality of such a set. Some standard graphs are used to illustrate the concept and determine this number. Bounds on γwtc are obtained for general graphs, and its relationship to other graph parameters are investigated. The paper aims to develop this new graph invariant and establish basic results about weak triple connected domination.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
This document discusses the concept of strong triple connected domination number (stc) of a graph. Some key points:
1. A subset S of vertices is a strong triple connected dominating set if S is a strong dominating set and the induced subgraph <S> is triple connected.
2. The strong triple connected domination number stc(G) is the minimum cardinality of a strong triple connected dominating set.
3. Some standard graphs for which the exact stc value is determined include paths, cycles, complete graphs, wheels, and more.
4. Bounds on stc(G) are established, such as 3 ≤ stc(G) ≤ p-1
This document provides information about graphs and graph theory concepts. It defines what a graph is consisting of vertices and edges. It describes different types of graphs such as undirected graphs, directed graphs, multigraphs, and pseudographs. It also discusses graph representations using adjacency matrices, adjacency lists, and incidence matrices. Additionally, it covers graph properties and concepts such as degrees of vertices, connected graphs, connected components, planar graphs, graph coloring, and the five color theorem.
This document provides an introduction to graph theory concepts. It defines what a graph is consisting of vertices and edges. It discusses different types of graphs like simple graphs, multigraphs, digraphs and their properties. It introduces concepts like degrees of vertices, handshaking lemma, planar graphs, Euler's formula, bipartite graphs and graph coloring. It provides examples of special graphs like complete graphs, cycles, wheels and hypercubes. It discusses applications of graphs in areas like job assignments and local area networks. The document also summarizes theorems regarding planar graphs like Kuratowski's theorem stating conditions for a graph to be non-planar.
ON ALGORITHMIC PROBLEMS CONCERNING GRAPHS OF HIGHER DEGREE OF SYMMETRYFransiskeran
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry.
On algorithmic problems concerning graphs of higher degree of symmetrygraphhoc
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that
Aut GR = G depends very much on the description of the geometry with which one starts. For example, we
can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded
subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3
special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form.
The aim of this research is to develop an effective method for generation of graphs connected with classical
geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The
main approach is to avoid manual drawing and to calculate graph layout automatically according to its
formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it
becomes more complicated for graphs of geometrical nature. There are two main reasons for the
investigations of spectra: (1) very often spectra carry much more useful information about the graph than a
corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called
Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important
for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one
could find some new Ramanujan graphs.
International Journal of Computational Engineering Research(IJCER) ijceronline
nternational Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
This document discusses various graph parameters related to distance between vertices in a graph, including diameter, radius, and average distance. It examines how these parameters are defined, how they relate to each other and other graph properties, and how they behave in different graph classes. It also discusses characterizations of graph classes based on distance properties and generalizations to concepts like Steiner trees.
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1. International Mathematical Forum, Vol. 7, 2012, no. 25, 1209 - 1222
Spanning Tree Algorithm for Families
of Chained Graphs
D. Lotfi, M. El Marraki and D. Aboutajdine
LRIT Unit´e associ´ee au CNRST(URAC29)- Faculty of Sciences
University of Mohammed V-Agdal, B.P.1014 RP, Rabat, Morocco
doun.lotfi@gmail.com, marraki@fsr.ac.ma, aboutaj@fsr.ac.ma
Abstract
Enumeration of trees is a new line of research in graph theory; many
researchers worked on this area, starting with the Matrix Tree Theorem
given by Kirchhoff who defined the number of spanning trees in graph
G as the determinant of its Laplacian matrix, since this later is easy to
compute but it cannot give the recurrences of spanning trees. In this
paper, we present two different methods to derive recursive functions,
the deletion and contraction method and the splitting method that cal-
culate the number of spanning trees in some families of graph, many
recursions are given for the wheel graph, fan graph and family of corn
graphs. The purpose of this paper is to propose an algorithm that cal-
culates the number of spanning trees in a chained graph. Finally, we
give the complexity of the chained wheel graph and the chained corn
graph as applications of the spanning trees algorithm.
Mathematics Subject Classification: 05C85, 05C30
Keywords: planar graph, complexity, spanning tree, grid graph, wheel
graph, fan graph
1 Introduction
The most classical theories of interest concerning spanning trees is the com-
plexity or the number of spanning trees of a given graph. Kirchhoff [7] gave
a theorem called the Matrix Tree Theorem that determinates the number of
spanning trees in a graph G as the value of any cofactor of the matrix DG −AG
such that DG is the degree matrix ( A diagonal matrix DG = diag(d1, ..., dn)
corresponding to the graph that has the vertex degree of di in the ith entry)
and AG is the adjacency matrix of G (Boolean matrix such that the position
aij is equal to the number of edges between the ith vertex and jth vertex).
2. 1210 D. Lotfi, M. El Marraki and D. Aboutajdine
The disadvantage of this method is that it can not produce the recurrence that
gives the sequence of numbers. The motivation in this paper is the following
classes of graph:
• Fan graph Fn,
• Wheel graph Wn,
• Corn graph Nn, with n is the number of petals.
All these classes of graphs can formed which we called the k chained graph,
a chain that contains k graph mentioned above. For each of these families of
graph, an explicit recurrence is given and its corresponding algorithm to count
the number of spanning trees.
2 History and outline
The first historical result is the Cayley’s Theorem [3], which gives an explicit
formula to count the number of spanning trees of a planar graph C. Later
in 2008, Bogdanowicz [1] gave a formula for the number of spanning trees in
a Fan graph. In 2009, Haghighi and Bibak [6] used the Cayley’s Theorem to
find a recursive function that calculates the number of spanning trees in Fan
and Wheel graph. Recently, El Marraki, Lotfi and Modabish [8],[12] gave a
new method to find explicit recursions counting the number of spanning trees
for the planar graph case. In the latest paper and this one, the main objective
is the same: Our aim is to find a recursive function to count the number of
spanning trees for some families of graph. The first result in this paper is to
give recursive functions that calculates the complexity of chained wheel graph
and chained corn graph. As a consequence, we propose an algorithm counting
the two complexities which is based on calculating the power of a specific
matrix.
3 Basic Vocabulary
We shall use the standard terminology of graph theory, as it is introduced in
most books on the theory of graphs [2], [5], [13]. In this paper, a graph implies
a finite simple graph which has neither loops or multiple edges.
Definition 3.1 A graph is said to be planar if it can be illustrated in the
plane so that arcs representing edges meet only at points representing vertices.
Informally, G is planar if it can be drawn in the plane without any edge cross-
ings. It can also be called as a map.
3. Spanning tree algorithm for families of chained graphs 1211
A graph G can have many planar drawing, it depends on the visual field
through we can observe the graph. The following figure illustrates a planar
embedding of a graph G.
Figure 1: Planar embedding of a graph
Euler [13] gave a formula that relates the number of vertices, edges and
faces of a planar graph:|VM| + |FM| − |EM| = 2. (eg.,The previous planar
graph (See Figure 1) has 7 vertices, 8 edges and 3 faces).
Definition 3.2 Spanning Tree of undirected connected graph G is a tree
formed by all the vertices and some or all the edges of G.
The following figure gives some spanning trees of a graph G.
Figure 2: Some spanning trees of graph G
Definition 3.3 Let G be a graph, the complexity of G is the number of
trees spanning G, it is denoted by τ(G) and it depends mostly on the number
of edges in G (eg., the complexity of the graph mentioned above (See Figure 2)
is 8).
Throughout the paper, a graph implies a finite connected indirected graph,
the connectivity is that there must be a path between each pair of vertices.
Note that the exact geometric positions of vertices or the lengths of the edges
are not important. We will find many recursions in this paper, all concerning
the spanning trees in the fan graph, wheel graph and corn graph.
4 Spanning trees Recursions
The first recursive formula that calculates the number of spanning trees in
graph G was given by Cayley [3] as a special case. Note that Cayley’s formula
count the number of trees spanning the graph G that satisfies the criteria of
planarity, connectivity and disorientation.
4. 1212 D. Lotfi, M. El Marraki and D. Aboutajdine
4.1 Spanning trees recursions by deletion and contrac-
tion
As we mentioned before, a connected graph may have an exponential number
of spanning trees by deleting some edges and preserving connectivity. We shall
now define the proprieties of deletion and contraction an edge, the deletion is
the graph (V, E − e) and G/e is the contraction obtained from G − e by fusing
two vertices v1 and v2 (See Figure 3).
Figure 3: Graph G, G − e and G/e
Theorem 4.1 Let G be a graph, e ∈ G then the spanning trees of G that
don’t contain the edge e and the spanning trees of the deletion G − e are in
bijection, alternatively, the spanning trees of G that contain e and the spanning
trees of the contraction G/e are in bijection.
Let T ∈ τ(G), {T ∈ τ(G) | e ∈ T} ←→ τ(G−e) and {T ∈ τ(G) | e ∈
T} ←→ τ(G/e).
For any graph G and edge e, the deletion and contraction recursion that
was proved by Kirchhoff [7] is given by the following formula:
τ(G) = τ(G − e) + τ(G/e)
4.1.1 Complexity of the corn graph
Let Np be the corn graph that contains p + 2 vertices with p is the number of
petals as illustrated in the following figure 4 (a).
Lemma 4.2 The complexity of the corn graph Np is given by the following
formula
τ(Np) =
1
√
5
(
φ3
− φp+2
1 − φ
−
¯φ3
− ¯φp+2
1 − ¯φ
) + 5, p ≥ 2
with φ = 3+
√
5
2
and ¯φ = 3−
√
5
2
.
5. Spanning tree algorithm for families of chained graphs 1213
Figure 4: Corn Graphs Np, Np and Np
Proof 4.3 Let Np be the corn graph and Fp be the fan graph (See figure 6),
from Theorem 4.1 we get
τ(Np) = τ(Fp) + τ(Np−1)
= τ(Fp) + τ(Fp−1) + ... + τ(F4) + τ(F3) + τ(F2) + τ(N1)
=
1
√
5
((φp+1
+ φp
+ φp−1
+ ... + φ2
) − (¯φp+1
+ ¯φp
+ ¯φp−1
+ ... + ¯φ2
)) + 5
=
1
√
5
(
φ3
− φp+2
1 − φ
−
¯φ3
− ¯φp+2
1 − ¯φ
) + 5, p ≥ 2, φ =
3 +
√
5
2
and ¯φ =
3 −
√
5
2
.
Let Np be the corn graph obtained by deleting the last edge from the graph
Np (See figure 4 (b)).
Lemma 4.4 The complexity of the corn graph Np = Np − e is given by the
following formula:
τ(Np) = τ(Fp−1) + τ(Np−1), p ≥ 2.
Let Np be the corn graph obtained by deleting the first and the last edges
from the corn graph Np (See figure 4 (c)).
Lemma 4.5 The complexity of the corn graph Np is given by the following
formula:
τ(Np ) = τ(Fp−2) + τ(Np−1), p ≥ 2.s
The proof of the Lemma 4.4 and 4.5 is similar to the previous one. Note
that we use this result in section 6 to calculate the number of spanning trees
in the chained corn graph.
6. 1214 D. Lotfi, M. El Marraki and D. Aboutajdine
4.2 Spanning trees recursions by splitting
Definition 4.6 Let G has a separation pair x, y (an edge e = xy or a simple
path p = x, a, b, ...., y such that all of its vertices have degree 2 except x and y)
then we can split G into two graphs G1 and G2, called split graphs, we shall
denote G = G1 ‡ G2 such that G1 is obtained from G1 by adding a new simple
path p = x, a, b, ..., y, similarly, we define the split graph G2 (See figure 5) [13].
Figure 5: Graph G = G1 ‡ G2
Theorem 4.7 Let G be a planar graph that can be split in two graphs G1
and G2, G = G1 ‡ G2 such that ‡ is a simple path that contains k edges (See
figure 5), the number of spanning trees in the graph G is given by the following
equation:
τ(G) = τ(G1) × τ(G2) − k2
τ(G1) × τ(G2) [11]
Examples: Let Wp+q+1 be the wheel graph defined by p + q petals and
contains p + q + 1 vertices, 2(p + q) edges and p + q + 1 faces, and let Fp+1 be
the fan graph that contains p−1 petals, p+1 vertices, 2p−1 edges and p faces
(See figure 6). In [12] and [14], the authors give the complexity of the wheel
and fan graphs by using the deletion and contraction method or the splitting
method studied above.
The number of spanning trees in Wp+q+1 and Fp+1 is given by the following
equations:
τ(Wp+q+1) = (3+
√
5
2
)p+q
+ (3−
√
5
2
)p+q
− 2, p, q ≥ 2.
τ(Fp+1) = 1√
5
((3+
√
5
2
)p−1
− (3−
√
5
2
)p−1
), p ≥ 2.
7. Spanning tree algorithm for families of chained graphs 1215
Figure 6: Wheel and Fan graphs
5 Spanning Tree Algorithm
For the results above, it was necessary to use the spanning tree recursion to
find the complexity of a graph G, however, for the family of the chained graphs,
the formula that gives the number of spanning tree is complex. In this section,
we give an algorithm that enumerates the number of spanning trees for some
families of chained graphs.
Definition 5.1 Let Ci be a planar graph, we call Gn as a chained graph if it
is a succession of n split graphs of type Ci, we denote Gn = Ci ‡ Ci ‡ Ci ‡ ... ‡ Ci
such that ‡ is a simple path that contains k edges (See figure 7).
Figure 7: Chained planar graph Gn
Theorem 5.2 Let Gn be a planar chained graph of type Gn = Ci ‡Ci ‡Ci ‡...‡
Ci such that ‡ is a simple path that contains k+1 vertices such that deg(vj) = 2
for j = 2, 3, ..., k and k edges, the number of spanning tree in the graph Gn is
given by the following system
8. 1216 D. Lotfi, M. El Marraki and D. Aboutajdine
τ(Gn) = τ(Gn−1) × τ(G1) − k2
τ(Gn−1) × τ(G1)
τ(Gn) = τ(Gn−1) × τ(G1) − k2
τ(Gn−1) × τ(G1 ), n ≥ 2
Where Gn is the planar chained graph obtained by deleting the last path p and
Gn is the planar chained graph obtained by deleting the first and last path p
(See figure 7).
Proof 5.3 Let Gn be the planar chained graph (See figure 7), by using the
splitting method (Theorem 4.3) we get,
therefore, τ(Gn) = τ(Gn−1) × τ(G1) − k2
τ(Gn−1) × τ(G1), the same proof
goes for Gn, we use the splitting method then we get the second equation of the
system.
Corollary 5.4 Let Gn be a planar chained graph of type Gn = Ci ‡Ci ‡...‡Ci
such that ‡ is a simple path that contains k + 1 vertices such that deg(vj) = 2
for j = 2, 3, ..., k and k edges, and Gn is the planar graph obtained after deleting
the last path p from the graph Gn, we denote gn = τ(Gn) and gn = τ(Gn), the
number of spanning tree in the graphs Gn and Gn is given by the following
equation
gn
gn
= Mn−1 g1
g1
, where M =
g1 -k2
g1
g1 -k2
g1
Example: Let Rn be the grid chained graph, it contains 2n + 2 vertices
(See Figure 8).
Figure 8: Grid Chained graph Rn
9. Spanning tree algorithm for families of chained graphs 1217
The complexity of Rn is given by the following equation:
τ(Rn) = 1
2
√
3
((2 +
√
3)n+1
− (2 −
√
3)n+1
) n ≥ 1 [4], [9], [10].
Since the complexity can be easy to compute by using the previous formula,
we can also use the Theorem 5.2 then we get the following system:
τ(Rn) = 4τ(Rn−1) − τ(Rn−1)
τ(Rn) = 4τ(Rn−1), n ≥ 2
The complexity matrix is defined by the initials conditions r1, r1 and r1 ,
M =
4 -1
4 0
then using the previous theorem, the complexity of the grid chained graph
is based on the power of the complexity matrix and the initial conditions r1 and
r1. To obtain the complexity of the grid chained graph, we run the following
algorithm:
1. Let n = 1.
2. Let R1 and R1 be the initial conditions of the grid chained graph.
3. Let V the vector formed by the initial conditions V =< r1, r1 >.
4. Let M the complexity matrix formed by r1, r1 and r1 .
5. For a given n, Using the square and multiply method, we calculate Mn
.
6. Calculate the product Mn
× V .
6 Applications
6.1 Complexity of the chained wheel graph
Let En be the chained wheel graph, it is formed by a chain of wheels that
contains p + q + 1 vertices, and let En be the chained Fan graph obtained by
deleting the last edge of each wheel from the chained wheel graph (See Figure
9).
10. 1218 D. Lotfi, M. El Marraki and D. Aboutajdine
Figure 9: The chained Wheel graph En
Theorem 6.1 Let En be the chained Wheel graph and En be the chained
Fan graph, the complexity of En and En are given by the following system:
τ(En) = τ(En−1) × τ(Wp+q+1) − τ(En−1) × τ(Fp+q+1)
τ(En) = τ(En−1) × τ(Fp+q+1) − τ(En−1) × τ(Fp+1) × τ(Fq+1)
with τ(E1) = τ(Wp+q+1) and τ(E1) = τ(Fp+q+1), p, q ≥ 1, n ≥ 1.
Proof 6.2 The graphs En and En belong to the family of chained graph Gn,
then the complexities τ(En) and τ(En) can be found by using the Theorem 5.2,
where the simple path p is the path that connects the last wheel with the chained
wheel En−1 .
The spanning trees recursions of the graph En and En can be found by
using the result in the Theorem 6.1, the downside of this method is that it
can produce a huge function which is not easy to compute, the algorithm given
earlier can formulates the complexity system by the following matrix equation:
Let en, en, wp+q+1, fp+q+1, fp+1 and fq+1 be respectively the complexities of the
graphs En, En, Wp+q+1, Fp+q+1, Fp+1 and Fq+1 .
en
en
= M
en−1
en−1
, where M =
wp+q+1 - fp+q+1
fp+q+1 -fp+1 × fq+1
en
en
= M
en−1
en−1
= ... = Mn−1 e1
e1
For a given n, we calculate the power n of the complexity matrix and the
multiplication with the vector formed by the initial conditions gives the com-
plexity of the wheel chained graph. Let En be the chained wheel graph, it is
11. Spanning tree algorithm for families of chained graphs 1219
formed by n wheels of type Wp+q+1 such that p = 3 and q = 3 (See figure 9),
the complexity of En in this case is given by the following equation:
en
en
= Mn−1 320
144
, where M =
320 -144
144 -64
The following table gives some values of the number of spanning trees in
the chained wheel graph En.
n τ(En)
1 320
2 81664
3 20824064
4 5310054400
5 1354042966016
6 345275625373696
7 88043925096366080
8 22450854264574050304
9 5724879446906287161344
10 1459821719716278556426240
Table 1: Some values of the complexity τ(En).
Our algorithm does not give only the complexity of the chained wheel
graph, it gives also the number of spanning trees in the chained fan graph En.
Another example of a chained graph is given in the following section, there-
fore, we can keep the same basic method used to calculate the number of span-
ning trees in the chained wheel graph. Note that we use the Lemma 4.2, 4.4
and 4.5 to calculate the number of spanning trees in the corn graphs which are
the initial conditions of our algorithm.
6.2 Complexity of the chained Corn graph
The chained corn graph On is the juxtaposition of n corns, each one contains
p + 2 vertices such that p is the number of petals (Triangles) as illustrated in
figure 10.
Theorem 6.3 Let τ(On) be the complexity of the chained corn graph of
type Np and τ(On) be the complexity of the chained corn graph of type Np, we
have the following system:
τ(On) = τ(On−1) × τ(O1) − τ(On−1) × τ(O1)
τ(On) = τ(On−1) × τ(O1) − τ(On−1) × τ(O1 )
τ(O1) = τ(Np), τ(O1) = τ(Np) and τ(O1 ) = τ(Np ), n ≥ 1, p ≥ 2.
12. 1220 D. Lotfi, M. El Marraki and D. Aboutajdine
Figure 10: The chained Corn graph On
Proof 6.4 From the Theorem 5.2, we obtain the system that calculates the
complexity of the chained corn graph.
The number of spanning trees in the chained corn graph can be found by
using the Theorem 6.3 or the algorithm given above. Let on, on, np, np and np
be respectively the complexities of the graphs On, On, Np, Np and Np .
on
on
= M
on−1
on−1
, where M =
np - np
np -np
on
on
= M
on−1
on−1
= ... = Mn−1 o1
o1
Let On be the chained corn graph, it is formed by n corns of type N3, each
one has 3 petals (see Figure 10), the complexity of On in this case is given by
the following equation:
on
on
= Mn−1 34
21
, where M =
34 -21
21 -11
The following table gives some values of the number of spanning trees in
the chained corn graph On of type N3.
n 1 2 3 4 5 6 7
τ(On) 34 715 14167 277936 5443339 106575085 2086523242
Table 2: Some values of the complexity τ(On).
7 Conclusion
Many applications in computer science such as data structure, algorithms,
compilers and expression evaluation require a tree representation, one of the
huge targets of these applications is to find a recursive function that calcu-
lates the number of spanning trees in graph G, however, for some families of
13. Spanning tree algorithm for families of chained graphs 1221
graph, the spanning tree recursions can be complex. In this paper, we sur-
vey the several methods that can derive spanning tree recursions such as the
deletion and contraction method and the splitting method, therefore, we gave
the complexity of the families of corn graph, moreover, we gave an algorithm
that counts the number of spanning trees in some families of graph without
using recursive functions, finally, as applications we gave the complexity of the
chained wheel graph and the chained corn graph. Our future purpose takes two
different ways, the first one is the generalization of the splitting method, such
that the decomposition of the planar graph can not be necessarily through a
simple path and as consequence we will give a general method that can derive
recursive functions for another families of planar graphs such as the grid graph
Gn,m. The second direction is the generalization of the spanning tree algorithm
given above such that it will count the complexity of any planar graph.
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Received: October, 2011