The document discusses characteristics of (γ, 3)-critical graphs. It begins by providing examples of (γ, 3)-critical graphs, such as the circulant graph C12 1, 4 and the Cartesian product Kt Kt . It then shows that a (γ, k)-critical graph is not necessarily (γ, k′)-critical for k ≠ k′ between 1 and 3. The document also verifies properties of (γ, 3)-critical graphs, such as not having vertices of degree 3. It concludes by proving characteristics about (γ, 3)-critical graphs that are paths, including that they have no vertices in V+ and satisfy other properties.
A labeling of graph G is a mapping that carries a set of graph elements into a set of numbers (Usually positive integers) called labels. An edge magic labeling on a graph with p vertices and q edges will be defined as a one-to-one map taking the vertices and edges onto the integers 1,2,----,
p+q with the property that the sum of the label on an edge and the labels of its end vertices is constant independent of the choice of edge.
In [1] Abdel-Aal has introduced the notions of m-shadow graphs and n-splitting graphs, for all m, n ³ 1.
In this paper, we prove that, the m-shadow graphs for paths and complete bipartite graphs are odd
harmonious graphs for allm³ 1. Also, we prove the n-splitting graphs for paths, stars and symmetric
product between paths and null graphs are odd harmonious graphs for all n³ 1. In addition, we present
some examples to illustrate the proposed theories. Moreover, we show that some families of graphs admit
odd harmonious libeling.
DISTANCE TWO LABELING FOR MULTI-STOREY GRAPHSgraphhoc
An L (2, 1)-labeling of a graph G (also called distance two labeling) is a function f from the vertex set V (G) to the non negative integers {0,1,…, k }such that |f(x)-f(y)| ≥2 if d(x, y) =1 and | f(x)- f(y)| ≥1 if d(x, y) =2. The L (2, 1)-labeling number λ (G) or span of G is the smallest k such that there is a f with
max {f (v) : vє V(G)}= k. In this paper we introduce a new type of graph called multi-storey graph. The distance two labeling of multi-storey of path, cycle, Star graph, Grid, Planar graph with maximal edges and its span value is determined. Further maximum upper bound span value for Multi-storey of simple
graph are discussed.
In this paper, we show that the number of edges for any odd harmonious Eulerian graph is congruent to 0 or 2 (mod 4), and we found a counter example for the inverse of this statement is not true. We also proved that, the graphs which are constructed by two copies of even cycle Cn sharing a common edge are odd harmonious. In addition, we obtained an odd harmonious labeling for the graphs which are constructed by two copies of cycle Cn sharing a common vertex when n is congruent to 0 (mod 4). Moreover, we show that, the Cartesian product of cycle graph Cm and path Pn for each n ≥ 2, m ≡ 0 (mod 4) are odd harmonious graphs. Finally many new families of odd harmonious graphs are introduced.
A labeling of graph G is a mapping that carries a set of graph elements into a set of numbers (Usually positive integers) called labels. An edge magic labeling on a graph with p vertices and q edges will be defined as a one-to-one map taking the vertices and edges onto the integers 1,2,----,
p+q with the property that the sum of the label on an edge and the labels of its end vertices is constant independent of the choice of edge.
In [1] Abdel-Aal has introduced the notions of m-shadow graphs and n-splitting graphs, for all m, n ³ 1.
In this paper, we prove that, the m-shadow graphs for paths and complete bipartite graphs are odd
harmonious graphs for allm³ 1. Also, we prove the n-splitting graphs for paths, stars and symmetric
product between paths and null graphs are odd harmonious graphs for all n³ 1. In addition, we present
some examples to illustrate the proposed theories. Moreover, we show that some families of graphs admit
odd harmonious libeling.
DISTANCE TWO LABELING FOR MULTI-STOREY GRAPHSgraphhoc
An L (2, 1)-labeling of a graph G (also called distance two labeling) is a function f from the vertex set V (G) to the non negative integers {0,1,…, k }such that |f(x)-f(y)| ≥2 if d(x, y) =1 and | f(x)- f(y)| ≥1 if d(x, y) =2. The L (2, 1)-labeling number λ (G) or span of G is the smallest k such that there is a f with
max {f (v) : vє V(G)}= k. In this paper we introduce a new type of graph called multi-storey graph. The distance two labeling of multi-storey of path, cycle, Star graph, Grid, Planar graph with maximal edges and its span value is determined. Further maximum upper bound span value for Multi-storey of simple
graph are discussed.
In this paper, we show that the number of edges for any odd harmonious Eulerian graph is congruent to 0 or 2 (mod 4), and we found a counter example for the inverse of this statement is not true. We also proved that, the graphs which are constructed by two copies of even cycle Cn sharing a common edge are odd harmonious. In addition, we obtained an odd harmonious labeling for the graphs which are constructed by two copies of cycle Cn sharing a common vertex when n is congruent to 0 (mod 4). Moreover, we show that, the Cartesian product of cycle graph Cm and path Pn for each n ≥ 2, m ≡ 0 (mod 4) are odd harmonious graphs. Finally many new families of odd harmonious graphs are introduced.
A Note on Non Split Locating Equitable Dominationijdmtaiir
Let G = (V,E) be a simple, undirected, finite
nontrivial graph. A non empty set DV of vertices in a graph
G is a dominating set if every vertex in V-D is adjacent to
some vertex in D. The domination number (G) of G is the
minimum cardinality of a dominating set. A dominating set D
is called a non split locating equitable dominating set if for
any two vertices u,wV-D, N(u)D N(w)D,
N(u)D=N(w)D and the induced sub graph V-D is
connected.The minimum cardinality of a non split locating
equitable dominating set is called the non split locating
equitable domination number of G and is denoted by nsle(G).
In this paper, bounds for nsle(G) and exact values for some
particular classes of graphs were found.
In this paper, we introduce the notions of m-shadow graphs and n-splitting graphs,m ³ 2, n ³ 1. We
prove that, the m-shadow graphs for paths, complete bipartite graphs and symmetric product between
paths and null graphs are odd graceful. In addition, we show that, the m-splitting graphs for paths, stars
and symmetric product between paths and null graphs are odd graceful. Finally, we present some examples
to illustrate the proposed theories.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
E-Cordial Labeling of Some Mirror GraphsWaqas Tariq
Let G be a bipartite graph with a partite sets V1 and V2 and G\' be the copy of G with corresponding partite sets V1\' and V2\' . The mirror graph M(G) of G is obtained from G and G\' by joining each vertex of V2 to its corresponding vertex in V2\' by an edge. Here we investigate E-cordial labeling of some mirror graphs. We prove that the mirror graphs of even cycle Cn, even path Pn and hypercube Qk are E-cordial graphs.
Strong (Weak) Triple Connected Domination Number of a Fuzzy Graphijceronline
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.
It is a algorithm used to find a minimum cost spanning tree for connected weighted undirected graph.This algorithm first appeared in Proceedings of the American Mathematical Society in 1956, and was written by Joseph Kruskal.
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.
A Note on Non Split Locating Equitable Dominationijdmtaiir
Let G = (V,E) be a simple, undirected, finite
nontrivial graph. A non empty set DV of vertices in a graph
G is a dominating set if every vertex in V-D is adjacent to
some vertex in D. The domination number (G) of G is the
minimum cardinality of a dominating set. A dominating set D
is called a non split locating equitable dominating set if for
any two vertices u,wV-D, N(u)D N(w)D,
N(u)D=N(w)D and the induced sub graph V-D is
connected.The minimum cardinality of a non split locating
equitable dominating set is called the non split locating
equitable domination number of G and is denoted by nsle(G).
In this paper, bounds for nsle(G) and exact values for some
particular classes of graphs were found.
In this paper, we introduce the notions of m-shadow graphs and n-splitting graphs,m ³ 2, n ³ 1. We
prove that, the m-shadow graphs for paths, complete bipartite graphs and symmetric product between
paths and null graphs are odd graceful. In addition, we show that, the m-splitting graphs for paths, stars
and symmetric product between paths and null graphs are odd graceful. Finally, we present some examples
to illustrate the proposed theories.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
E-Cordial Labeling of Some Mirror GraphsWaqas Tariq
Let G be a bipartite graph with a partite sets V1 and V2 and G\' be the copy of G with corresponding partite sets V1\' and V2\' . The mirror graph M(G) of G is obtained from G and G\' by joining each vertex of V2 to its corresponding vertex in V2\' by an edge. Here we investigate E-cordial labeling of some mirror graphs. We prove that the mirror graphs of even cycle Cn, even path Pn and hypercube Qk are E-cordial graphs.
Strong (Weak) Triple Connected Domination Number of a Fuzzy Graphijceronline
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.
It is a algorithm used to find a minimum cost spanning tree for connected weighted undirected graph.This algorithm first appeared in Proceedings of the American Mathematical Society in 1956, and was written by Joseph Kruskal.
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.
A total dominating set D of graph G = (V, E) is a total strong split dominating set if the induced subgraph < V-D > is totally disconnected with atleast two vertices. The total strong split domination number γtss(G) is the minimum cardinality of a total strong split dominating set. In this paper, we characterize total strong split dominating sets and obtain the exact values of γtss(G) for some graphs. Also some inequalities of γtss(G) are established.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
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.
New Classes of Odd Graceful Graphs - M. E. Abdel-AalGiselleginaGloria
In this paper, we introduce the notions of m-shadow graphs and n-splitting graphs,m ≥ 2, n ≥ 1. We
prove that, the m-shadow graphs for paths, complete bipartite graphs and symmetric product between
paths and null graphs are odd graceful. In addition, we show that, the m-splitting graphs for paths, stars
and symmetric product between paths and null graphs are odd graceful. Finally, we present some examples
to illustrate the proposed theories.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal,
On the Equality of the Grundy Numbers of a Graphjosephjonse
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.
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.
Abstract - Let G = (V,E) be a simple, undirected, finite nontrivial graph. A non empty set DÍV of vertices in a graph G is a dominating set if every vertex in V-D is adjacent to some vertex in D. The domination number g(G) of G is the minimum cardinality of a dominating set. A dominating set D is called a non split locating equitable dominating set if for any two vertices u,wÎV-D, N(u)ÇD ¹ N(w)ÇD, ½N(u)ÇD½=½N(w)ÇD½ and the induced sub graph áV-Dñ is connected.The minimum cardinality of a non split locating equitable dominating set is called the non split locating equitable domination number of G and is denoted by gnsle(G). In this paper, bounds for gnsle(G) and exact values for some particular classes of graphs were found.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Welocme to ViralQR, your best QR code generator.ViralQR
Welcome to ViralQR, your best QR code generator available on the market!
At ViralQR, we design static and dynamic QR codes. Our mission is to make business operations easier and customer engagement more powerful through the use of QR technology. Be it a small-scale business or a huge enterprise, our easy-to-use platform provides multiple choices that can be tailored according to your company's branding and marketing strategies.
Our Vision
We are here to make the process of creating QR codes easy and smooth, thus enhancing customer interaction and making business more fluid. We very strongly believe in the ability of QR codes to change the world for businesses in their interaction with customers and are set on making that technology accessible and usable far and wide.
Our Achievements
Ever since its inception, we have successfully served many clients by offering QR codes in their marketing, service delivery, and collection of feedback across various industries. Our platform has been recognized for its ease of use and amazing features, which helped a business to make QR codes.
Our Services
At ViralQR, here is a comprehensive suite of services that caters to your very needs:
Static QR Codes: Create free static QR codes. These QR codes are able to store significant information such as URLs, vCards, plain text, emails and SMS, Wi-Fi credentials, and Bitcoin addresses.
Dynamic QR codes: These also have all the advanced features but are subscription-based. They can directly link to PDF files, images, micro-landing pages, social accounts, review forms, business pages, and applications. In addition, they can be branded with CTAs, frames, patterns, colors, and logos to enhance your branding.
Pricing and Packages
Additionally, there is a 14-day free offer to ViralQR, which is an exceptional opportunity for new users to take a feel of this platform. One can easily subscribe from there and experience the full dynamic of using QR codes. The subscription plans are not only meant for business; they are priced very flexibly so that literally every business could afford to benefit from our service.
Why choose us?
ViralQR will provide services for marketing, advertising, catering, retail, and the like. The QR codes can be posted on fliers, packaging, merchandise, and banners, as well as to substitute for cash and cards in a restaurant or coffee shop. With QR codes integrated into your business, improve customer engagement and streamline operations.
Comprehensive Analytics
Subscribers of ViralQR receive detailed analytics and tracking tools in light of having a view of the core values of QR code performance. Our analytics dashboard shows aggregate views and unique views, as well as detailed information about each impression, including time, device, browser, and estimated location by city and country.
So, thank you for choosing ViralQR; we have an offer of nothing but the best in terms of QR code services to meet business diversity!
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
The Art of the Pitch: WordPress Relationships and Sales
1452 86301000013 m
1. Applicable Analysis and Discrete Mathematics
available online at http://pefmath.etf.rs
Appl. Anal. Discrete Math. 4 (2010), 197–206.
doi:10.2298/AADM100206013M
CHARACTERISTICS OF (γ, 3)-CRITICAL GRAPHS
D. A. Mojdeh, P. Firoozi
In this note the (γ, 3)-critical graphs are fairly classified. We show that
a (γ, k)-critical graph is not necessarily a (γ, k )-critical for k = k and
k, k ∈ {1, 2, 3}. The (2, 3)-critical graphs are definitely characterized. Also
the properties of (γ, 3)-critical graphs are verified once their edge connectivity
are 3.
1. INTRODUCTION
Let G = (V, E) be a graph with vertex set V of order n and edge set E. A set
S ⊆ V is a dominating set if every vertex in V is either in S or is adjacent to a vertex
in S. The domination number γ(G) is the minimum cardinality of a dominating set
of G, and a dominating set of minimum cardinality is called a γ(G)-set. Note
that removing a vertex can increase the domination number by more than one,
but can decrease it by at most one. We define a graph G to be (γ, k)-critical, if
γ(G − S) < γ(G) for any set S of k vertices [1]. Obviously, a (γ, k)-critical graph
G has γ(G) ≥ 2, unless in trivial case k = |V (G)| that γ = 1 (for more, we refer to
[1–5]).
The open neighborhood of a vertex v ∈ V is N (v) = {x ∈ V |vx ∈ E} while
N [v] = N (v) ∪ {v} is the closed neighborhood. So a set S ⊆ V is a dominating set,
if V = s∈S N [s]. The connectivity of G, written κ(G), is the minimum size of a
vertex set S such that G − S is disconnected or has only one vertex. A graph G
is k-connected if its connectivity is at least k. A graph is k-edge-connected if every
disconnecting set has at least k edges. The edge connectivity of G, written λ(G),
is the minimum size of a disconnecting set. We denote the distance between two
vertices x and y in G by dG (x, y) and the minimum degree of G with δ(G), the
pendant vertex is a vertex of degree 1 and the support vertex is a vertex adjacent
2000 Mathematics Subject Classification. 05C69.
Keywords and Phrases. Vertex domination number, (γ, 3)-critical graph, edge connectivity.
197
2. 198
D. A. Mojdeh, P. Firoozi
to a pendant vertex. A graph G is vertex-transitive if for every pair u, v ∈ V (G)
there is an automorphism that maps u to v. In a graph G = (V, E), if |V | = n then
we say G is of order n. Let Pn , Cn and Kn be a path, a cycle and a complete graph
of order n respectively ( for more, we refer to [6]).
In this note, the characteristics of (γ, 3)-critical graphs are studied and we
shall know the individualities of (γ, 3)-critical graphs.
The following result is useful.
Observation A. ([2], Observation 5) If G is any graph and x, y ∈ V (G) such that
γ(G − {x, y}) = γ(G) − 2, then dG (x, y) ≥ 3.
2. EXAMPLES OF (γ, 3)-CRITICAL GRAPHS
In this section, we present two examples of (γ, 3)-critical graphs: circulant graph C12 1, 4 and the Cartesian product Kt Kt . In general, the circulant graph Cn+1 1, m is a graph with vertex set {v0 , v1 , . . . , vn } and edge set
{vi vi+j (mod n+1) | i ∈ {0, 1, . . . , n} and j ∈ {1, m}}.
The graph Gt = Kt Kt is the Cartesian product of complete graph Kt by
itself. The graph Gt has t disjoint copies of Kt in rows and t disjoint copies of Kt
in columns. For ease of discussion, we will use the words row and column to mean
a ”copy of Kt ”. The vertices of ith row are vi1 , vi2 , · · · , vit and the vertices of j th
column are v1j , v2j , · · · , vtj for 1 ≤ i, j ≤ t.
Proposition 1. The circulant C12 1, 4 is (4, 1)-critical, (4, 2)-critical and (4, 3)critical.
Proof. Let G = C12 1, 4 . It has domination number 4, and {v0 , v3 , v6 , v9 } is
a minimum dominating set for C12 1, 4 . The set vertices {v3 , v6 , v9 } dominates
G − {v0 }. Since G is vertex transitive, then G is (4, 1)-critical. In what follows,
S G means that the set S dominates G.
For (4, 2)-criticality, {v3 , v6 , v9 } G − {v0 , v1 } and G − {v0 , v5 }, {v3 , v7 , v9 }
G−{v0 , v2 }, {v5 , v7 , v10 } G−{v0 , v3 }, {v2 , v7 , v9 } G−{v0 , v4 } and {v3 , v9 }
G − {v0 , v6 }, now the vertex transitivity of G prove that G is (4, 2)-critical.
For (4, 3)-criticality, {v5 , v7 , v10 }
G − {v0 , v1 , v2 }, {v6 , v8 , v11 }
G−
{v0 , v1 , v3 }, {v6 , v7 , v9 }, G−{v0 , v1 , v4 }, {v3 , v8 , v10 } G−{v0 , v1 , v5 }, {v3 , v9 }
G − {v0 , v1 , v6 }, {v7 , v9 , v10 },
G − {v0 , v2 , v4 }, {v4 , v7 , v9 },
G − {v0 , v2 , v5 },
{v3 , v9 }
G − {v0 , v2 , v6 }, {v4 , v5 , v10 }
G − {v0 , v2 , v7 }, {v8 , v9 , v10 }
G−
{v0 , v3 , v6 }, {v5 , v8 , v10 } G−{v0 , v3 , v7 }, {v2 , v7 , v9 } G−{v0 , v4 , v6 }, {v2 , v6 , v10 }
G − {v0 , v4 , v8 }, {v3 , v9 } G − {v0 , v5 , v6 }, here the vertex transitivity of G implies that G is (4, 3)-critical too.
Proposition 2. The graph Gt = Kt Kt for t ≥ 3 is (t, 3)-critical.
Proof. By removing three vertices vij , vsr , vkl from Gt , there are three cases:
suppose i = s = k. Without loss of generality, let these three vertices be v11 , v12 , v13 .
3. Characteristics of (γ, 3)-critical graphs
199
Then {vss | 2 ≤ s ≤ t} is a dominating set of cardinality t − 1. Suppose s = i and
i = k. Without loss of generality, let the vertices be v11 , v12 , v33 , then {v23 , v32 } ∪
{vss | 4 ≤ s ≤ t} is a dominating set of cardinality t − 1. Suppose s, i and k are
mutually distinct. Without loss of generality let the vertices be v11 , v22 , v33 , then
{v32 , v23 } ∪ {vss | 4 ≤ s ≤ t} is a dominating set of cardinality t − 1. Thus, for three
vertices u, v and w of Gt , γ(Gt − {u, v, w}) ≤ t − 1 implying Gt is (t, 3)-critical.
3. (γ, k) AND (γ, k )-CRITICALITY FOR 1 ≤ k = k ≤ 3
In the following examples we show (γ, k)-critical graphs are not necessarily
(γ, k )-critical graphs for 1 ≤ k = k ≤ 3.
(1) The cycle C3n+1 is a (n + 1, 1)-critical graph but is not (n + 1, k)-critical for
k ∈ {2, 3}.
(2) Let G = (V, E), x ∈ V and G[x] be a graph with vertex set V ∪ {x } and edge
set E ∪ {x y : y ∈ NG [x]}, thus G[x] is obtained from G by adding a new vertex x
that has the same closed neighborhood as x. Let G be the circulant graph C 8 1, 4
with vertex set {v0 , v1 , . . . , v7 }, then G[v0 ] is (3, 2)-critical but is not (3, k)-critical
for k ∈ {1, 3} (Figure 1 (a)).
(3) Let H be a graph constructed from the Cartesian product K3 K3 by adding a
new vertex x adjacent to v11 , v12 , v23 and v33 . Let H[x] be a graph constructed from
H using same method in (2). It is easy to see that H[x] (Figure 1 (b)) is (3, 3)critical but is not (3, k)-critical for k ∈ {1, 2}. Also the path P4 is a (2, 3)-critical
but is not (2, k)-critical for k ∈ {1, 2}.
(4) The circulant C8 1, 4 is (3, k)-critical for k ∈ {1, 2} but is not (3, 3)-critical.
(5) Let G = K2n − M where M is a perfect matchings of K2n . Graph G is (2, k)critical for k ∈ {1, 3} but is not (2, 2)-critical (see Proposition 9). For n = 5 see
the Figure 1(c) K10 − M where M = {v1 v2 , v3 v4 , v5 v6 , v7 v8 , v9 v10 } which is not
(2, 2)-critical, because of γ(G − {v1 , v2 }) = 2.
(6) The graph H (see (3), Figure 1 (b) once x is omitted) is (3, k)-critical for
k ∈ {2, 3} but is not (3, 1)-critical (because of γ(H − {x}) = 3).
(7) The Harary graph H2m,n(2m+1)+2m (m ≥ 2) by making each vertex adjacent
to the nearest m vertices in each direction around the circle (see Figure 1 (d) for
m = 2 = n and more generally, we refer to [6]), is not (γ, k)-critical, for k ∈ {1, 2, 3}
because of its domination number is n + 1 and each vertex just dominates 2m + 1
vertices [5]. But γ(H2m,n(2m+1)+2m − {v1 }) = γ(H2m,n(2m+1)+2m − {v1 , v2 }) =
γ(H2m,n(2m+1)+2m − {v1 , v2 , v3 }) = n + 1 = γ(H2m,n(2m+1)+2m ).
(8) C12 1, 4 and Gt = Kt Kt are (γ, k)-critical for k ∈ {1, 2, 3}. (See Propositions
1, 2 and also Proposition 2 of [2]).
4. 200
D. A. Mojdeh, P. Firoozi
Figure 1.
4. CHARACTERISTICS OF (γ, 3)-CRITICAL GRAPHS
By noting that removing three vertices can decrease the domination number,
we can prove some useful results.
Observation 3. For a (γ, 3)-critical graph G and x, y, z ∈ V (G), γ(G) − 3 ≤
γ(G − {x, y, z}) ≤ γ(G) − 1.
Observation 4. Let G be any graph and x1 , x2 , x3 ∈ V (G). If γ(G−{x1 , x2 , x3 }) =
γ(G) − 3. Then dG (xi , xj ) ≥ 3 for i = j.
Proof. On the contrary, suppose, without loss of generality, that D is a γ(G −
{x1 , x2 , x3 })-set and that dG (x1 , x2 ) ≤ 2. Let y be a common adjacent vertex (if
the distance is 2) or be x1 (if the distance is 1). Then D ∪ {y, x3 } dominates G and
so γ(G − {x1 , x2 , x3 }) < 3, which is a contradiction.
As an immediate result we have:
Observation 5. If G is a connected (γ, 3)-critical graph such that diam(G) = 2,
then ∀ x, y, z ∈ V (G), γ(G − {x, y, z}) ≥ γ(G) − 2.
Observation 4 implies that, if γ(G−{x, y, z}) = γ(G)−3 for any three distinct
vertices x, y and z, then G has no edge.
5. Characteristics of (γ, 3)-critical graphs
201
The proof of the result below has been given for any k in [4].
Observation 6. The (γ, 3)-critical graph does not have a vertex of degree 3.
Let V (G) = V 0 ∪ V + ∪ V − where V 0 = {v ∈ V |γ(G − v) = γ(G)}, V + =
{v ∈ V |γ(G − v) > γ(G)}, V − = {v ∈ V |γ(G − v) < γ(G)}.
Proposition 7. If G = P3 is a connected (γ, 3)-critical graph, then V = V − ∪ V 0 ,
that is, V + = ∅. Furthermore, (1) either G is (γ, 1)-critical, or G−v is (γ, 2)-critical
for all v ∈ V 0 and (2) either G is (γ, 2)-critical or G − {v, w} is (γ, 1)-critical for
every {v, w} such that γ(G − {v, w}) = γ(G).
Proof. Suppose that V + = ∅ and a ∈ V + (G), then γ(G − {a}) ≥ γ(G) + 1.
Since G is (γ, 3)-critical, then for all a, b, c ∈ V (G), γ(G − {a, b, c}) ≤ γ(G) − 1.
Furthermore γ(G − {a}) = γ(G) + 1, because, if γ(G − {a}) > γ(G) + 1, then it is
impossible for the removal of two vertices in V (G) − a to decrease the domination
number 3. So γ(G − {a}) = γ(G) + 1. Now γ((G − {a}) − {b, c}) = γ(G) − 1. So
dG−{a} (b, c) ≥ 3 (Observation A), that follows G − {a} has no edge, hence G is a
star. We claim that G has no edge. Since G = P3 is (γ, 3)-critical and a ∈ V + so G
is not a star of center a with degree at least 2. Now let ax be an edge and G = ax,
therefore γ(G − {a}) = γ(G), then a ∈ V + (G), a contradiction. So V + = ∅. Next
parts of proposition has straightforward proof and dispense with it.
(2, 3)-critical graphs are characterized.
Observation 8. There is no (2, 3)-critical graph of order 5.
Proof. Let G be a (2, 3)-critical graph and x, y, z be any three vertices of G. Then
G − {x, y, z} = K2 . It shows that G = K5 , a contradiction.
Proposition 9. A graph G is (2, 3)-critical if and only if G = P2 ∪ P1 , P4 , 2P2 ,
P3 ∪ P1 , C3 ∪ P1 or G = K2n − M where n ≥ 2 and M is a perfect matching of
K2n .
Proof. Let G = P2 ∪P1 , P4 , 2P2 , P3 ∪P1 , C3 ∪P1 . It is clear that G is (2, 3)-critical.
Let G = K2n − M where n ≥ 2 and M is a perfect matching of K2n . Then each
vertex is adjacent to 2n − 2 vertices and since n ≥ 2, deleting any 3 vertices of G
implies that there exists a vertex with degree 2n − 4. Thus γ(G − {x, y, z}) = 1.
Conversely, let G be a (2, 3)-critical graph. The (2, 3)-critical graphs with 4
vertices are P2 ∪ P1 , C4 = K4 − M, P4 , 2P2 , P3 ∪ P1 and C3 ∪ P1 . Observation
8 implies that, there is not (2, 3)-critical graph of order 5. Let V (G) ≥ 6. We
show that G is (γ, 1)-critical. If G is not (γ, 1)-critical, then there is a v ∈ V 0
such that G − v is (2, 2)-critical. If G − v is connected, then by Proposition 13
of [2] γ(G − v) ≥ 3 a contradiction. If G − v is disconnected, then there are two
vertices x, y in G − v such that G − {v, x, y} has two nonempty components. So
γ(G − {v, x, y}) ≥ 2 a contradiction. Thus G is (2, 1)-critical and the results of [1]
imply that G = K2n − M.
As an immediate result of Proposition 9 we have:
6. 202
D. A. Mojdeh, P. Firoozi
Corollary 10. If G is a connected (γ, 3)-critical graph with |V (G)| ≥ 6 and
G = K2n − M where M is a perfect matching, then γ(G) ≥ 3.
Remark 1.
1. Let G be a (γ, 3)-critical graph with a pendant vertex x and support vertex y, then
deg(y) = 2. Let u and w be the pendant vertex and support vertex respectively with
deg(w) ≥ 3. Vertices x, y are two neighborhoods of w other than u, then γ(G−{w, x, y}) =
γ(G) − 1, because u is an isolated vertex in (G − {w, x, y}), and then it belongs to a
γ(G−{w, x, y})-set, D. Now (D −{u})∪{w} is a γ(G)-set with the cardinality of γ(G)−1,
a contradiction.
2. Let G be any graph with a pendant vertex, then G is not (γ, 1)-critical and (γ, 2)critical. Because the support vertex w belongs to V 0 and γ(G − {w, v}) = γ(G) where v
is an adjacent vertex of w other than pendant vertex.
3. If G = P4 , C4 is a (γ, 3)-critical graph and is not (γ, 1)-critical , then G has at most
one vertex of degree 2. Suppose v ∈ V 0 and G has at least two vertices of degree 2
such as u, w with neighborhoods {x, y} and {z, t} respectively. Suppose that v ∈ {x, y},
then γ(G − {x, y, v}) = γ(G) − 1. Since the vertex u is an isolated vertex and belongs to
every γ(G − {x, y, v})-set, then γ(G) − 1 vertices dominate G − {v}. Hence v ∈ V − , a
contradiction.
4. Any (γ, 3)-critical graph G other than P3 and P4 has at most one pendant vertex.
Because more than one pendant vertex in G leads to at least 2 support vertices of degree
2.
Proposition 11. Let G be a connected (γ, 3)-critical graph. If G is a graph other
than P3 , C3 , P4 and C4 , then G has at most one vertex of degree 1, one vertex of
degree 2 and the other vertices of degree at least 4.
Proof. By Observation 6 and Remark 1 the result holds.
Remark 2. By Remark 1 and Proposition 11, one can say that, almost all (γ, 3)-critical
graphs have δ(G) ≥ 4.
The below result has been proved in [4]. For seeing result, the below definition
is added.
Definition. A block of a graph G is a maximal connected subgraph of G that has
no cut-vertex.
Corollary 12. A graph G is (γ, 1)-critical, (γ, 2)-critical and (γ, 3)-critical if and
only if each block of G is (γ, 1)-critical, (γ, 2)-critical and (γ, 3)-critical. Further,
if G is (γ, 1)-critical, (γ, 2)-critical and (γ, 3)-critical with blocks G 1 , G2 , . . . , Gk ,
k
then γ(G) =
i=1
γ(Gi ) − k + c(G), where c(G) is the number of components of G.
Now we find a (γ, 1)-critical, (γ, 2)-critical and (γ, 3)-critical graph Gγ with
given γ ≥ 3 and diameter γ − 1.
7. Characteristics of (γ, 3)-critical graphs
203
Proposition 13. For every integer γ ≥ 3, there exists a connected graph G γ
that is (γ, 1)-critical, (γ, 2)-critical and (γ, 3)-critical satisfying γ(G γ ) = γ and
diam(Gγ ) = γ − 1.
Proof. Let H be the Cartesian product K3 K3 . Then diam(H) = 2 and by
Proposition 2, H is (3, 1)-critical, (3, 2)-critical and (3, 3)-critical. Let F be the
circulant C12 1, 4 then diam(F )=3 and, by Proposition 1, F is (4, 1)-critical, (4, 2)critical and (4, 3)-critical. If γ = 3 or γ = 4, then we can take Gγ = H or Gγ = F,
respectively. Hence we may assume that γ ≥ 5. We consider two possibilities,
depending on whether γ is odd or even.
Suppose γ = 2k+1, where k ≥ 2. Let u and w be any two nonadjacent vertices
of H. Let B1 , B2 , . . . , Bk be k disjoint copies of H. For i = 1, 2 . . . , k, let ui and wi
denote the vertices of Bi corresponding to u and w, respectively in H. Let Gγ be
obtained by identifying wi and ui+1 for i = 1, 2, . . . , k − 1. Then B1 , B2 , . . . , Bk are
blocks of Gγ . Since each Bi is (γ, 1)-critical, (γ, 2)-critical and (γ, 3)-critical with
γ(Bi ) = 3, we know from Corollary 12, that Gγ is (γ, 1)-critical, (γ, 2)-critical and
(γ, 3)-critical with γ(Gγ ) = 2k + 1 = γ. Furthermore, diam(Gγ ) = 2k = γ − 1.
Suppose γ = 2k, where k ≥ 3. In the construction of Gγ in the preceding
paragraph, replace Bk−1 and Bk with a copy L of F. Then B1 , B2 , . . . , Bk−2 , L are
blocks of Gγ . By Corollary 12, Gγ is (γ, 1)-critical, (γ, 2)-critical and (γ, 3)-critical
with γ(Gγ ) = 2k = γ. Furthermore, diam(Gγ ) = 2k − 1 = γ − 1.
5. EDGE CONNECTIVITY
As it has been seen in the previous section, there exist connected (γ, 3)-critical
graphs that contain cut-vertices. In this section we study the edge connectivity
λ(G) of (γ, 3)-critical graphs.
In any graph a vertex of degree 1 leads to λ ≤ 1 and a vertex of degree 2
leads to λ ≤ 2. So by Observation 6 we have:
Observation 14. If G is a (γ, 3)-critical graph and λ(G) ≥ 3, then δ(G) ≥ 4.
Theorem 15. Suppose that G is a connected (γ, 3)-critical graph with λ(G) = 3
and an edge cut {ab, cd, ef }. Let G1 and G2 be two components of G − ab − cd − ef,
with a, c, e ∈ V (G1 ), b, d, f ∈ V (G2 ) and a, c, e are distinct. Then the following
must all be true.
(i) It is not the case that b = d = f. Hereafter b, d, f are distinct or (b = d) = f
(ii) γ(G) = γ(G1 ) + γ(G2 ), if (b = d) = f.
(iii) a, c, e ∈ V + (G1 ), if (b = d) = f.
(iv) If {a, c, e} ⊆ V − (G1 ), then none of b, d, f is in γ(G2 )-set.
(v) If b, d, f are distinct vertices, then γ(G2 − {b, d, f }) = γ2 − 1 and a γ(G2 −
{b, d, f })-set simultaneously dominates none of two of b, d, f.
8. 204
D. A. Mojdeh, P. Firoozi
(vi) If b, d, f are distinct and belong to V 0 (G2 ), then there is a γ(G2 − {b, d})-set
containing f, a γ(G2 − {b})-set containing d or a γ(G2 − {d})-set containing b.
(vii) If b, d, f are distinct. Then there is no γ(G1 )-set containing {a, c, e}.
(viii) Let b, d, f are distinct and {a, c, e} ⊆ V − (G1 ). There is no γ(G1 − {x})-set
containing {y, z}, where {x, y, z} = {a, c, e}.
Proof. Let γ(G) = γ. For i = 1, 2, let Vi = V (Gi ) and let γi = γ(Gi ).
(i) Let b = d = f. We show that, there is a γ(G2 )-set containing b. λ(G) = 3 implies
that deg(b) ≥ 6, in other word degG2 (b) ≥ 3. Thus there are at least two vertices
x, y in V (G2 ) such that b ∈ N (x) ∩ N (y), so γ(G2 − {x, b, y}) = γ(G2 ) − 1. Let D2
be a γ(G2 )-set to include b. Now γ − 1 ≥ γ(G − {a, c, e}) = γ(G1 − {a, c, e}) + γ2 ,
and so γ(G1 − {a, c, e}) ≤ γ1 − 1. Let D1 be a γ(G1 − {a, c, e})-set. Then D1 ∪ D2 is
a dominating set for G of cardinality |D1 ∪ D2 | ≤ γ − 1, a contradiction. Therefore
f = (b = d), b = (d = f ), d = (b = f ) or b, d, f are distinct.
(ii) Clearly, γ ≤ γ1 + γ2 . It suffices to show that γ ≥ γ1 + γ2 . Since (b = d) = f
and δ(G) ≥ 4, there is a vertex x = f such that x ∈ V (G2 ) ∩ N (b). It is clear
γ2 (G2 − {b, x, f }) ≤ γ2 − 2. Suppose that γ2 (G2 − {b, x, f }) = γ2 − 2, there is a γ2 set D2 for G2 includes b and f. Now, γ −1 ≥ γ(G−{a, c, e}) = γ(G1 −{a, c, e})+γ2 .
Let D1 be γ(G1 − {a, c, e})-set, hence |D1 ≤ γ1 − 1 and then D = D1 ∪ D2 is a
γ-set with cardinality |D| ≤ γ − 1, a contradiction. Thus γ2 (G2 − {b, x, f }) = γ2 − 1
and γ − 1 ≥ γ(G − {b, x, f }) = γ(G1 ) + γ2 (G2 − {b, x, f }) = γ1 + γ2 − 1. Therefore
γ ≥ γ 1 + γ2 .
(iii) Suppose (b = d) = f and a ∈ V + (G1 ). It is well known γ2 −2 ≤ γ(G2 −{b, f }) ≤
γ2 .
First, let γ(G2 −{b, f }) = γ2 −2. There is a γ2 -set D2 for G2 includes b and f.
Now, γ −1 ≥ γ(G−{a, c, e}) = γ(G1 −{a, c, e})+γ2 . Let D1 be γ(G1 −{a, c, e})-set,
hence |D1 ≤ γ1 − 1 and then D = D1 ∪ D2 is a γ-set of cardinality |D| ≤ γ − 1 a
contradiction.
Second, let γ2 − 1 ≤ γ(G2 − {b, f }) ≤ γ2 . Then γ(G2 − {b, f }) ≥ γ2 − 1 and
γ − 1 ≥ γ(G − {a, b, f }) = γ(G1 − {a}) + γ(G2 − {b, f }) =≥ γ1 + 1 + γ2 − 1 = γ, a
contradiction. Same proof can be used for c and e.
(iv) Suppose b is in γ(G2 ) − set = D2 . Let D1 be a γ(G1 − {a})-set. Since a ∈
V − (G1 ), |D1 | = γ1 −1. Now D = D1 ∪D2 dominates G and |D| = γ1 −1+γ2 = γ−1,
a contradiction. Hence b does not belong to any γ(G2 ) − set. The result for d and
f, follows from an identical argument.
(v) Since G is (γ, 3)-critical γ − 1 ≥ γ(G − {b, d, f }) = γ(G1 ) + γ(G2 − {b, d, f })
and γ(G2 − {b, d, f }) ≤ γ2 − 1. Let D2 be a γ(G2 − {b, d, f })-set, if |D2 | = γ2 − 3,
then D2 ∪ {b, d, f } is a γ(G2 )-set, a contradiction with (iv). If |D2 | = γ2 − 2, then
there are two vertices x, y that dominate b, d, f. If x dominates b, then D2 ∪ {b, y}
is a dominating set of G2 a contradiction, thus |D2 | = γ2 − 1. Now if D2 dominates
b and d, then D2 ∪ {f } is a γ(G2 )-set that also a contradiction.
9. Characteristics of (γ, 3)-critical graphs
205
(vi) The part (v) implies γ(G2 − {b, d, f }) = γ2 − 1. {b, d, f } ⊆ V 0 (G2 ) implies
γ2 = γ(G2 − {b}) = γ(G2 − {d}) = γ(G2 − {f }). There are two cases.
1. Let γ(G2 − {b, d}) = γ2 . Then γ(G2 − {b, d, f }) = γ2 − 1. Suppose that
D2 = γ(G2 − {b, d, f })-set, so D2 ∪ {f } is a γ(G2 − {b, d})-set.
2. Let γ(G2 − {b, d}) = γ2 − 1 and let D = γ(G2 − {b, d})-set. Then D ∪ {d} is
a γ(G2 − {b}-set and D ∪ {b} is a γ(G2 − {d}.
(vii) Suppose there is a γ(G1 )-set D1 containing a, c and e. Let D2 be a γ(G2 −
{b, d, f })-set. By (vi) |D2 | ≤ γ2 − 1, and so D1 ∪ D2 is a dominating set for G of
cardinality γ1 + γ2 − 1, a contradiction.
(viii) Suppose there is a γ(G1 − {a})-set D1 containing c, e, then D1 ∪ {a} =
γ(G1 )-set and there is a γ(G1 )-set containing a, c, e contradicting (vii). Identical
arguments show there is no a γ(G1 − {c})-set containing a, e, and there is no a
γ(G1 − {e})-set containing a, c.
Theorem 16. Let G be a connected graph. If G is (3, 3)-critical or (4, 3)-critical,
γ1 = γ2 and λ(G) = 1, 2 then λ(G) ≥ 4.
Proof. λ(G) = 1, 2 and (γ, 3)-criticality imply that δ(G) ≥ 4. Let λ(G) = 3 and
{ab, cd, ef } be an edge cut. Let G1 and G2 be two components of G − ab − cd − ef,
with a, c, e ∈ V (G1 ), b, d, f ∈ V (G2 ). By (ii) of Theorem 15, γ(G) = γ(G1 ) +
γ(G2 ). Let γ(G) = 3. Without loss of generality, suppose 2 ≤ γ(G1 ) ≤ 3 and
0 ≤ γ(G2 ) ≤ 1. Let γ(G1 ) ≥ 2 and γ(G2 ) = 1. There are at least 5 vertices in
G2 , because of δ(G) ≥ 4. Let {x, y, z} ⊆ V (G2 ) such that contains {b, d, f }, then
γ(G2 − {x, y, z}) = 1 − 1 = 0, a contradiction.
Let γ(G1 ) = 3 and γ(G2 ) = 0, so b, d, f all vertices of V (G2 ) should be
dominated by a, c, e. Since at least one of b, d and f of degree 2 or 3, in G2 that
also a contradiction. Therefore λ(G) ≥ 4.
Now let γ(G) = 4. Without loss of generality, suppose 3 ≤ γ(G1 ) ≤ 4 and
0 ≤ γ(G2 ) ≤ 1. These are proved by using manner of proof once γ(G) = 3, 2 ≤
γ(G1 ) ≤ 3 and 0 ≤ γ(G2 ) ≤ 1.
We close with some open questions.
Questions
1. Characterize the (γ, 3)-critical graphs.
2. Is it true that if G is a connected (γ, 3)-critical graph of order at least 6, then
λ(G) ≥ 3 and δ(G) ≥ 4? Though, we know that for (γ, 3)-critical graphs, if
λ(G) = 3, then δ(G) ≥ 4.
Acknowledgements. The authors would like to thank the referee(s) for useful
comments and valuable suggestions.
10. 206
D. A. Mojdeh, P. Firoozi
REFERENCES
1. R. C. Brigham, P. Z. Chinn, R. D. Dutton: Vertex domination critical graphs.
Networks, 18 (1988), 173–179.
2. R. C. Brigham, T. W. Haynes, M. A. Henning, D. F. Rall: Bicritical domination.
Discrete Mathematics, 305 (2005), 18–32.
3. T. W. Haynes, S. T. Hedetniemi, P. J. Slater: Fundamentals of Domination in
Graphs. Marcel Dekker, New York, 1998.
4. D. A. Mojdeh, P. Firoozi, R. Hasni: On connected (γ, k)-critical graphs. Australasian Journal of Combinatorics, 46 (2010), 25–35.
5. D. A. Mojdeh, R. Hasni: On questions on (total) domination vertex critical graphs.
To appear Ars combinatoria.
6. D. B. West: Introduction to Graph Theory (Second Edition). Prentice Hall USA
2001.
University of Tafresh, IRI,
Department of Mathematics,
University of Tafresh
E-mail: d.a.mojdeh@gmail.com
Department of Mathematics,
University of Mazandaran,
IRI
(Received July 27, 2009)
(Revised February 6, 2010)