Dominating Induced Matchings for P7-Free Graphs in Linear Time

📝 Abstract

Let $G$ be a finite undirected graph with edge set $E $. An edge set $E' \subseteq E$ is an {\em induced matching} in $G$ if the pairwise distance of the edges of $E'$ in $G$ is at least two; $E'$ is {\em dominating} in $G$ if every edge $e \in E \setminus E'$ intersects some edge in $E' $. The \emph{Dominating Induced Matching Problem} (\emph{DIM}, for short) asks for the existence of an induced matching $E'$ which is also dominating in $G $; this problem is also known as the \emph{Efficient Edge Domination} Problem. The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is \NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three. However, its complexity was open for $P_k $-free graphs for any $k \ge 5 $; $P_k$ denotes a chordless path with $k$ vertices and $k-1$ edges. We show in this paper that the weighted DIM problem is solvable in linear time for $P_7 $-free graphs in a robust way.

💡 Analysis

Let $G$ be a finite undirected graph with edge set $E $. An edge set $E' \subseteq E$ is an {\em induced matching} in $G$ if the pairwise distance of the edges of $E'$ in $G$ is at least two; $E'$ is {\em dominating} in $G$ if every edge $e \in E \setminus E'$ intersects some edge in $E' $. The \emph{Dominating Induced Matching Problem} (\emph{DIM}, for short) asks for the existence of an induced matching $E'$ which is also dominating in $G $; this problem is also known as the \emph{Efficient Edge Domination} Problem. The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is \NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three. However, its complexity was open for $P_k $-free graphs for any $k \ge 5 $; $P_k$ denotes a chordless path with $k$ vertices and $k-1$ edges. We show in this paper that the weighted DIM problem is solvable in linear time for $P_7 $-free graphs in a robust way.

📄 Content

arXiv:1106.2772v1 [cs.DM] 14 Jun 2011 Dominating Induced Matchings for P7-Free Graphs in Linear Time Andreas Brandst¨adt∗ Raffaele Mosca† June 4, 2018 Abstract Let G be a finite undirected graph with edge set E. An edge set E′ ⊆E is an induced matching in G if the pairwise distance of the edges of E′ in G is at least two; E′ is dominating in G if every edge e ∈E \ E′ intersects some edge in E′. The Dominating Induced Matching Problem (DIM, for short) asks for the existence of an induced matching E′ which is also dominating in G; this problem is also known as the Efficient Edge Domination Problem. The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three. However, its complexity was open for Pk-free graphs for any k ≥5; Pk denotes a chordless path with k vertices and k −1 edges. We show in this paper that the weighted DIM problem is solvable in linear time for P7-free graphs in a robust way. Keywords: dominating induced matching; efficient edge domination; P7-free graphs; linear time algorithm; robust algorithm. 1 Introduction Let G be a simple undirected graph with vertex set V and edge set E. A subset M of E is an induced matching in G if the G-distance of every pair of edges e, e′ ∈M, e ̸= e′, is at least two, i.e., e ∩e′ = ∅and there is no edge xy ∈E with x ∈e and y ∈e′. A subset M ⊆E is a dominating edge set if every edge e ∈E \ M shares an endpoint with some edge e′ ∈M, i.e., if e ∩e′ ̸= ∅. A dominating induced matching (d.i.m. for short) is an induced matching which is also a dominating edge set. Let us say that an edge e ∈E is matched by M if e ∈M or there is e′ ∈M with e∩e′ ̸= ∅. Thus, M is a d.i.m. of G if and only if every edge of G is matched by M but no edge is matched twice. The Dominating Induced Matching Problem (DIM, for short) asks whether a given graph has a dominating induced matching. This can also be seen as a special 3-colorability problem, namely the partition into three independent vertex sets A, B, C such that G[B ∪ ∗Fachbereich Informatik, Universit¨at Rostock, A.-Einstein-Str. 21, D-18051 Rostock, Germany, ab@informatik.uni-rostock.de †Dipartimento di Scienze, Universit´a degli Studi “G. D’Annunzio” Pescara 65121, Italy. r.mosca@unich.it 1 C] is an induced matching: If M ⊆E is a d.i.m. of G then the vertex set has the partition V = A ∪V (M) with independent vertex set A, and independent sets B, C with B ∪C = V (M). Dominating induced matchings are also called edge packings in some papers, and DIM is known as the Efficient Edge Domination Problem (EED for short). A brief history of EED as well as some applications in the fields of resource allocation, encoding theory and network routing are presented in [16] and [19]. Grinstead et al. [16] show that EED is NP-complete in general. It remains hard for bipartite graphs [21]. In particular, [20] shows the intractability of EED for planar bipartite graphs and [10] for very restricted bipartite graphs with maximum degree three (the restrictions are some forbidden subgraphs). In [4], it is shown that the problem remains NP-complete for planar bipartite graphs with maximum degree three but is solvable in polynomial time for hole-free graphs (which was an open problem in [20] and is still mentioned as an open problem in [9]; actually, [9, 20] mention that the complexity of DIM is an open problem for weakly chordal graphs which are a subclass of hole-free graphs). In [9], as another open problem, it is mentioned that for any k ≥5, the complexity of DIM is unknown for the class of Pk-free graphs. Note that the complexity of the related problems Maximum Independent Set and Maximum Induced Matching is unknown for P5-free graphs, and a lot of work has been done on subclasses of P5-free graphs. In this paper, we show that for P7-free graphs, DIM is solvable in linear time. Actually, we consider the edge-weighted optimization version of DIM, namely the Minimum Dominating Induced Matching Problem (MDIM), which asks for a dominating induced matching M in G = (V, E) of minimum weight with respect to some given weight function ω : E →R (if existent). For P5-free graphs, DIM is solvable in time O(n2) as a consequence of the fact that the clique-width of (P5,gem)-free graphs is bounded [5, 6] and a clique-width expression can be constructed in time O(n2) [3]. In [9], it is mentioned that DIM is expressible in a certain kind of Monadic Second Order Logic, and in [12], it was shown that such problems can be solved in linear time on any class of bounded clique-width assuming that the clique-width expressions are given or can be determined in the same time bound. It is well known that the clique-width of cographs (i.e., P4-free graphs) is at most two (and such clique-width expressions can be determined in linear time) and thus the DIM problem can be solved in linear time on cographs. In section 4 we give

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