1. 1 Graphs with Maximal Induced Matchings of the Same Size Ph. Baptiste 1, M. Kovalyov 2, Yu. Orlovich 3, F. Werner 4, I. Zverovich 3 1 Ecole Polytechnique, Palaiseau, France 2 National Academy of Sciences of Belarus, Minsk, Belarus 3 Belarusian State University, Minsk, Belarus 4 Otto-von-Guericke-University of Magdeburg, Germany INCOM 2012, Bucharest / Romania, May 23 - 25, 2012

2. 2 1.Well-Indumatched Graphs 2.Complexity of Recognizing Well-Indumatched Graphs 3.NP-Completeness Results for Well-Indumatched Graphs 4.Perfectly Well-Indumatched Graphs Outline of the Talk

3. 3 Let G be a graph with vertex set V(G) and edge set E(G). 1. Well-Indumatched Graphs  An induced matching M is maximal if no other induced matching in G contains M.  An induced matching M in G is a set M  E(G) such that (i) M is a matching in G (a set of pairwise non- adjacent edges), (ii) there is no edge in E(G) \ M connecting two edges of M.

4. 4 A maximal induced matching of size 1 A maximal induced matching of size 2

5. 5  A graph G is called well-indumatched if all maximal induced matchings in G have the same size. For example, the graph S n obtained from a star K 1,n by subdividing each edge of K 1,n by two vertices is a well-indumatched graph. S3S3

6. 6  Let IMatch(G) be the set of all maximal induced matchings of graph G.  Define the minimum maximal and maximum induced matching numbers, respectively, as follows: and  (G) = max {|M| : M  IMatch(G)}.  (G) = min {|M| : M  IMatch(G)}

7. 7  (G) = 2,  (G) = 3 In a greedy way we can find both  (G) and  (G) in any well- indumatched graph G. It is well known that the decision analogue of the problem of computing  (G) is NP-complete (Stockmeyer and Vazirani, 1982; Cameron, 1989).

8. 8  A matching of a graph G is a set of edges in G with no common end-vertices. The problem of graph recognition, in which all maximal matchings have the same size, was first considered by Lesk et al. (1984). Graphs which satisfy this property are known in the literature as equimatchable. Lesk et al. (1984) showed that there exists a polynomial time algorithm which decides whether a given input graph is equimatchable.

9. 9 2. Complexity of Recognizing Well-Indumatched Graphs NON-WELL-INDUMATCHED GRAPHS Instance: A graph G. Question: Are there two maximal induced matchings M and N in G with |M|  |N|? We consider the following decision problem.

10. 10 Theorem 1. NON-WELL-INDUMATCHED GRAPHS is an NP-complete problem. (Proof is done by a reduction from 3-SATISFIABILITY.) Thus, it is unlikely that there exists a characterization of well-indumatched graphs which provides its polynomial recognition.

11. 11 Corollary 1. NON-WELL-INDUMATCHED GRAPHS is an NP-complete problem even for bi-size indumatched graphs.  A graph G is said to be bi-size indumatched if there exists an integer k ≥ 1 such that |M| ∈ {k, k + 1} for every maximal induced matching M in G. Theorem 1 implies the following interesting corollaries. Corollary 2. NON-WELL-INDUMATCHED GRAPHS is an NP-complete problem even for (2P 5,K 1,5 )-free graphs. Corollary 3. The decision problem corresponding to the problem of computing  (G) is NP-complete within bi-size indumatched graphs.

12. 12 Theorem 2. For any positive integer t, the problem of recognizing the class WIM(t) is co-NP-complete even for (2P 5,K 1,5 )-free graphs.  Let WIM(t) be the class of graphs having maximal induced matchings of at most t sizes. Note that, if t = 1, then WIM(1) is the class of well-indumatched graphs.

13. 13 3. NP-Completeness Results for Well-Indumatched Graphs  A set S  V(G) is called an independent set if no two vertices in S are adjacent.  A set D ⊆ V(G) is a dominating set if each vertex in V(G) \ D is adjacent to a vertex of D.  A set I ⊆ V(G) is called an independent dominating set if I is an independent set and I is a dominating set.

14. 14  The minimum cardinality of an independent dominating set of G is the independent domination number, and it is denoted by i (G).  The independence number of G, denoted by  (G), is the maximum cardinality of an independent set in G.  The minimum cardinality of a dominating set in G is the domination number of G, denoted by γ(G).

15. 15 INDEPENDENT SET Instance: A graph G and an integer k. Question: Is  (G)  k ? DOMINATING SET Instance: A graph G and an integer k. Question: Is γ(G)  k ? INDEPENDENT DOMINATING SET Instance: A graph G and an integer k. Question: Is i (G)  k ? The following three decision problems are known to be NP-complete:

16. 16 Theorem 3. INDEPENDENT SET, DOMINATING SET and INDEPENDENT DOMINATING SET are NP-complete problems for well-indumatched graphs. PARTITION INTO SUBGRAPHS P 3 Instance: A graph G with |V(G)| = 3q. Question: Does G have a partition into subgraphs P 3, i.e., is there a partition V 1  V 2  …  V q of V (G) such that G(V i ) contains a subgraph isomorphic to P 3 for all i = 1, 2, …, q ?

17. 17 Theorem 4. PARTITIONS INTO SUBGRAPHS P3 is NP-complete for well-indumatched graphs. Corollary 4. Computing  for Hamiltonian line graphs L(G) is NP-hard even if G is a well-indumatched graph. Corollary 5. NON-WELL-INDUMATCHED GRAPHS is an NP-complete problem even for (2P 5,K 1,5 )-free graphs.

18. 18 4. Perfectly Well-Indumatched Graphs  A graph G is perfectly well-indumatched if every induced subgraph of G is well-indumatched. Perfectly well-indumatched graphs constitute a hereditary subclass of the well-indumatched graphs. We characterize perfectly well-indumatched graphs in terms of forbidden induced subgraphs.

19. 19 Theorem 5. For a graph G, the following statements are equivalent: (i) G is a perfectly well-indumatched graph. (ii) G is a (P 5, kite, butterfly)-free graph. P5P5 kite butterfly

20. 20 Corollary 7. DOMINATING SET is an NP-complete problem even for perfectly indumatched graphs. Corollary 6. The class of well-indumatched graphs is polynomially recognizable. Theorem 6. The INDEPENDENT SET problem and the INDEPENDENT DOMINATING SET problem can be solved in polynomial time for perfectly well-indumatched graphs, even in their weighted versions.