Graphs with Maximal Induced Matchings of the Same Size

1. Graphs with Maximal InducedMatchings of the Same Size 1Ecole Polytechnique, Palaiseau, France 2National Academy of Sciences of Belarus, Minsk, Belarus 3Belarusian State University, Minsk, Belarus 4Otto-von-Guericke-University of Magdeburg, Germany Ph. Baptiste1, M. Kovalyov2, Yu. Orlovich3, F. Werner4, I. Zverovich3 INCOM 2012, Bucharest / Romania, May 23 - 25, 2012

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

3. 1. Well-Indumatched Graphs Let Gbe a graph with vertex set V(G) and edge set E(G). • An induced matchingM 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. • An induced matching M is maximal if no other induced matching in G contains M.

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

5. A graph G is called well-indumatched if all maximal induced matchings in G have the same size. S3 For example, the graph Sn obtained from a star K1,n by subdividing each edge of K1,n by two vertices is a well-indumatched graph.

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: (G) = min{|M| : M  IMatch(G)} and (G) = max{|M| : M  IMatch(G)}.

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

10. Theorem 1.NON-WELL-INDUMATCHED GRAPHS isan 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. 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 1.NON-WELL-INDUMATCHED GRAPHSis an NP-complete problem even for bi-size indumatched graphs. Corollary 2.NON-WELL-INDUMATCHED GRAPHSis an NP-complete problem even for(2P5,K1,5)-free graphs. Corollary 3.The decision problem corresponding to the problem of computing(G)is NP-complete within bi-size indumatched graphs.

12. 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. Theorem 2.For any positive integer t, the problem of recognizing the class WIM(t)is co-NP-complete even for (2P5,K1,5)-free graphs.

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 dominatingset 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 Iis anindependent set and I is a dominating set.

14. 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 numberofG, denoted by γ(G). • The minimum cardinality of an independent dominating • set of G is the independent domination number, and it • is denoted byi(G).

15. The following three decision problems are known to be NP-complete: 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 ?

16. Theorem 3.INDEPENDENT SET, DOMINATING SET and INDEPENDENT DOMINATING SET are NP-complete problems for well-indumatched graphs. PARTITION INTO SUBGRAPHS P3 Instance: A graph G with |V(G)| = 3q. Question: Does G have a partition into subgraphsP3, i.e., is there a partition V1V2 …  Vq of V (G) such that G(Vi) contains a subgraph isomorphic to P3 for all i= 1, 2, …, q ?

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 GRAPHSis an NP-complete problem even for(2P5,K1,5)-free graphs.

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. Theorem 5.For a graph G, the following statements are equivalent: (i) G is a perfectly well-indumatched graph. (ii) G is a (P5, kite, butterfly)-free graph. butterfly kite P5

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