The document summarizes an algorithm for finding an independent dominating set of minimum cardinality in P2 + P3-free graphs in polynomial time. It first discusses related work showing the problem is NP-complete for more general graph classes. It then presents an O(n5) time algorithm that works by generating subsets S of the vertex set and using these to recursively construct candidate independent dominating sets. Key lemmas prove the algorithm's correctness and polynomial runtime.

1. Independent domination in finitely
defined classes of graphs:
Polynomial algorithms
Vadim Lozin, Raffaele Mosca,
Christopher Purcell
Discrete Applied Mathematics 182
(2015) 2–14
報告者: 陳政謙

2. Outline
• Related works
• Independent domination in P2 + P3-free graphs
• References

3. Related works
Journal Topic Author Result
Theoretical Computer
Science 301 (2003)
271 – 284
Independent
domination in finitely
defined classes of
graphs
R. Boliac, V. Lozin Some sufficient
conditions for the
independent
domination problem
are to be NP-hard in a
finitely defined class
of graphs.
Information
Processing Letters 36
(1990) 231-236
Domination in convex
and chordal bipartite
graphs
Peter Damaschke,
Haiko Müller, Dieter
Kratsch
It is shown by a
reduction from 3SAT
that independent
dominating set
remains NP-complete
when restricted to
chordal bipartite
graphs.

4. Related works
Journal Topic Author Result
Operations Research
Letters 1 (1982) 134-
138
Independent
domination in
chordal graphs
Martin Farber There is a linear
algorithm to solve the
independent
domination problem
in chordal graphs.
Discrete Applied
Mathematics 143
(2004) 351 – 352
The weighted
independent
domination problem
is NP-complete
for chordal graphs
Gerard J. Chang This paper shows that
the weighted
independent
domination problem
is NP-complete for
chordal graphs.

5. Related works
Journal Topic Author Result
Discrete Mathematics
73 (1989) 249-260
On diameters and rad
ii of bridged graphs
Martin Farber This paper proved
that 2K2-free graphs
have polynomially
many maximal
independent sets.

6. The independent dominating set

7. The independent domination
problem

8. Pn graphs
P2
P3
P2 + P3

9. P2 + P3-free graphs

10. P2 + P3-free graphs

11. Algorithm: Generation-1
S = {{v1}}
u({v1}) = v2
v1 v3 v4 v5v2

12. Algorithm: Generation-1
S = {{v1}}
u({v1}) = v2
v1 v3 v4 v5v2

13. Algorithm: Generation-1
u({v2}) = v1
S = {{v1}, {v2}}
u({v1}) = v2
v1 v4 v5v2 v3

14. Algorithm: Generation-1
v1 v4 v5v2 v3
u({v2}) = v1
S = {{v1}, {v2}}
u({v1}) = v2

15. Algorithm: Generation-1
v1 v4 v5v2 v3
u({v2}) = v1
S = {{v1, v3}, {v2}}
u({v1, v3}) = v2

16. Algorithm: Generation-1
v1 v4 v5v2 v3
u({v2}) = v1
S = {{v1, v3}, {v2}}
u({v1, v3}) = v2

17. Algorithm: Generation-1
u({v3}) = v2
v1 v5v2 v3
u({v2}) = v1
S = {{v1, v3}, {v2}, {v3}}
u({v1, v3}) = v2
v4

18. Algorithm: Generation-1
v1 v5v2 v3 v4
u({v3}) = v2u({v2}) = v1
S = {{v1, v3}, {v2}, {v3}}
u({v1, v3}) = v2

19. Algorithm: Generation-1
v1 v5v2 v3 v4
u({v3}) = v2u({v2, v4}) = v1
S = {{v1, v3}, {v2, v4}, {v3}}
u({v1, v3}) = v2

20. Algorithm: Generation-1
v1 v5v2 v3 v4
u({v3}) = v2u({v2, v4}) = v1
S = {{v1, v3}, {v2, v4}, {v3}}
u({v1, v3}) = v2

21. Algorithm: Generation-1
v1 v5v2 v3 v4
u({v3}) = v2u({v2, v4}) = v1
S = {{v1, v3}, {v2, v4}, {v3}, {v1, v4}}
u({v1, v3}) = v2 u({v1, v4}) = v3

22. Algorithm: Generation-1
v1 v2 v3 v4
u({v3}) = v2u({v2, v4}) = v3
S = {{v1, v3}, {v2, v4}, {v3}, {v1, v4}}
u({v1, v3}) = v2 u({v1, v4}) = v3
v5

23. Algorithm: Generation-1
v1 v2 v3 v4
u({v3}) = v2u({v2, v4}) = v3
S = {{v1, v3}, {v2, v4}, {v3}, {v1, v4}}
u({v1, v3}) = v2 u({v1, v4}) = v3
v5

24. Algorithm: Generation-1
v1 v2 v3 v4
u({v3}) = v2u({v2, v4}) = v3
S = {{v1, v3, v5}, {v2, v4}, {v3}, {v1, v4}}
u({v1, v3, v5}) = v2 u({v1, v4}) = v3
v5

25. Algorithm: Generation-1
v1 v2 v3 v4
u({v3, v5}) = v2u({v2, v4}) = v3
S = {{v1, v3, v5}, {v2, v4}, {v3, v5}, {v1, v4}}
u({v1, v3, v5}) = v2
u({v1, v4}) = v3
v5

26. Algorithm: Generation-1
v1 v2 v3 v4
u({v3, v5}) = v2u({v2, v4}) = v3
S = {{v1, v3, v5}, {v2, v4}, {v3, v5}, {v1, v4}}
u({v1, v3, v5}) = v2
u({v1, v4}) = v3
v5

27. Algorithm: Generation-1
v1 v2 v3 v4
u({v3, v5}) = v2u({v2, v4}) = v3
S = {{v1, v3, v5}, {v2, v4}, {v3, v5}, {v1, v4}, {v1, v2, v5}}
u({v1, v3, v5}) = v2
u({v1, v4}) = v3
v5
u({v1, v2 , v5}) = v4

28. Algorithm: Generation-1
v1 v2 v3 v4
u({v3, v5}) = v2u({v2, v4}) = v3
S = {{v1, v3, v5}, {v2, v4}, {v3, v5}, {v1, v4}, {v1, v2, v5}}
u({v1, v3, v5}) = v4
u({v1, v4}) = v3
v5
u({v1, v2 , v5}) = v4

29. Lemma 3
• For a graph G with n vertices, Algorithm Generation-1 runs in
time O(n5) and the family S produced by this algorithm
contains O(n3) subsets of V(G).

30. If a set H ∈ S was created in Step 2.2
u
G
v
AG({v, u, w})
w
H := {v, w} ∪ AG ({v, u, w})

31. If a set H ∈ S was created in Step 2.1
G
v
AG({v, u})
u
We denote W the set of neighbors of u each of which is not dominated
by every maximal independent set in G[H].
W
H := {v} ∪ AG ({v, u})

32. Proposition 4
• A set H created in Step 2.1 contains an independent set
dominating G if and only if H – H0 contains an independent set
dominating W.

33. Lemma 5
• If ab is an edge in G[H – H0], then W ⊆ N(a) ∪ N(b).
G
v
AG({v, u})
u
W
a b

34. The proof of Lemma 5
• If ab is an edge in G[H – H0], then W ⊆ N(a) ∪ N(b).
G
v
AG({v, u})
u
W
a b

35. The partition of cliques in G[H – H0]
• Lemma 5 shows that if Q = {q1, ... , qp} is a component (clique)
in G[H – H0] and Wi = W ∩ AG(qi), then {W1, ... , Wp} is a
partition of W. We denote this partition by P(Q).
q1
q2 q3
v1
v2
v3
W
P(Q) = {{v1, v2}, {v3}}
Q

36. Lemma 6
• The set H – H0 contains a maximal independent set
dominating W if and only if there is an element (Y1, . . . , Yt) ∈
P(Q1) × … × P(Qt) such that Y1 ∩ … ∩ Yt = ∅.
P(Q1) = {{v1}, {v2, v3}}
P(Q2) = {{v1, v2}, {v3}}
{v1} ∩ {v3} = ∅

37. The proof of Lemma 6
G
v
H – H0
u
W
Y1
Y2
Q1
Q2

38. The proof of Lemma 6
G
v
H – H0
u
WY1
Q1
Q2
Y2

39. The proof of Lemma 6
• Therefore, I dominates W if and only if Y1 ∪ …
∪ Yt = W.
• By De Morgan’s law, this holds if and only if Y1
∩ … ∩ Yt = ∅.

40. Lemma 7
• Given a set W of n elements and a number of partitions
P1, . . . ,Pt of W, one can check if there is an element (Y1, . . . ,
Yt) ∈ P1 × …× Pt such Y1 ∩…∩ Yt = ∅ in O(n2) time.
P1
P2
{v1} {v2, v3}
W = {v1, v2, v3}
P1 = {{v1}, {v2, v3}}
P2 = {{v1, v2}, {v3}}
{v1, v2, v3}
{v1} {∅} {v2} {v3}

41. Theorem 8
• Given a P2 + P3-free graph G with n vertices, one can find an
independent dominating set of minimum cardinality in G in
O(n5) time.
By Lemma 3, the time complexity of Algorithm Generation-1 is
O(n5).
By Proposition 4, Lemmas 6 and 7, the problem of determining if
H ∈ S contains a maximal independent set dominating G can be
solved in O(n2) time.
Therefore, an independent dominating set of minimum
cardinality in G can be found in O(n5) time.

42. References