Polynomial-time Algorithms for Weighted Efficient Domination Problems in AT-free Graphs and Dually Chordal Graphs

An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the vertex set of the graph. The minimum weight efficient domination problem is the problem of finding an efficient dominating set of minimum weight in a given vertex-weighted graph; the maximum weight efficient domination problem is defined similarly. We develop a framework for solving the weighted efficient domination problems based on a reduction to the maximum weight independent set problem in the square of the input graph. Using this approach, we improve on several previous results from the literature by deriving polynomial-time algorithms for the weighted efficient domination problems in the classes of dually chordal and AT-free graphs. In particular, this answers a question by Lu and Tang regarding the complexity of the minimum weight efficient domination problem in strongly chordal graphs

Weighted Efficient Domination in Classes of P6P_6-free Graphs

In a graph $G$, an efficient dominating set is a subset $D$ of vertices such that $D$ is an independent set and each vertex outside $D$ has exactly one neighbor in $D$. The Minimum Weight Efficient Dominating Set (Min-WED) problem asks for an efficient dominating set of total minimum weight in a given vertex-weighted graph; the Maximum Weight Efficient Dominating Set (Max-WED) problem is defined similarly. The Min-WED/Max-WED is known to be $NP$-complete for $P_7$-free graphs, and is known to be polynomial time solvable for $P_5$-free graphs. However, the computational complexity of the Min-WED/Max-WED problem is unknown for $P_6$-free graphs. In this paper, we show that the Min-WED/Max-WED problem can be solved in polynomial time for two subclasses of $P_6$-free graphs, namely for ($P_6,S_{1,1,3}$)-free graphs, and for ($P_6$, bull)-free graphs

On Efficient Domination for Some Classes of HH-Free Bipartite Graphs

A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (\emph{e.d.s.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.s.\ in $G$, is known to be \NP-complete even for very restricted $H$-free graph classes such as for $2P_3$-free chordal graphs while it is solvable in polynomial time for $P_6$-free graphs. Here we focus on $H$-free bipartite graphs: We show that (weighted) ED can be solved in polynomial time for $H$-free bipartite graphs when $H$ is $P_7$ or $\ell P_4$ for fixed $\ell$, and similarly for $P_9$-free bipartite graphs with vertex degree at most 3, and when $H$ is $S_{2,2,4}$. Moreover, we show that ED is \NP-complete for bipartite graphs with diameter at most 6.Comment: arXiv admin note: text overlap with arXiv:1701.0341

New polynomial case for efficient domination in P6P_6-free graphs

In a graph $G$, an {\it efficient dominating set} is a subset $D$ of vertices such that $D$ is an independent set and each vertex outside $D$ has exactly one neighbor in $D$. The {\textsc{Efficient Dominating Set}} problem (EDS) asks for the existence of an efficient dominating set in a given graph $G$. The EDS is known to be $NP$-complete for $P_7$-free graphs, and is known to be polynomial time solvable for $P_5$-free graphs. However, the computational complexity of the EDS problem is unknown for $P_6$-free graphs. In this paper, we show that the EDS problem can be solved in polynomial time for a subclass of $P_6$-free graphs, namely ($P_6$, banner)-free graphs

Weighted Efficient Domination for (P5+kP2)(P_5+kP_2)-Free Graphs in Polynomial Time

Let $G$ be a finite undirected graph. A vertex {\em dominates} itself and all its neighbors in $G$. A vertex set $D$ is an {\em efficient dominating set} (\emph{e.d.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.\ in $G$, is known to be \NP-complete even for very restricted graph classes such as for claw-free graphs, for chordal graphs and for $2P_3$-free graphs (and thus, for $P_7$-free graphs). We call a graph $F$ a {\em linear forest} if $F$ is cycle- and claw-free, i.e., its components are paths. Thus, the ED problem remains \NP-complete for $F$-free graphs, whenever $F$ is not a linear forest. Let WED denote the vertex-weighted version of the ED problem asking for an e.d. of minimum weight if one exists. In this paper, we show that WED is solvable in polynomial time for $(P_5+kP_2)$-free graphs for every fixed $k$, which solves an open problem, and, using modular decomposition, we improve known time bounds for WED on $(P_4+P_2)$-free graphs, $(P_6,S_{1,2,2})$-free graphs, and on $(2P_3,S_{1,2,2})$-free graphs and simplify proofs. For $F$-free graphs, the only remaining open case is WED on $P_6$-free graphs

Weighted Efficient Domination for P6P_6-Free Graphs in Polynomial Time

In a finite undirected graph $G=(V,E)$, a vertex $v \in V$ {\em dominates} itself and its neighbors in $G$. A vertex set $D \subseteq V$ is an {\em efficient dominating set} ({\em e.d.} for short) of $G$ if every $v \in V$ is dominated in $G$ by exactly one vertex of $D$. The {\em Efficient Domination} (ED) problem, which asks for the existence of an e.d. in $G$, is known to be NP-complete for $P_7$-free graphs but solvable in polynomial time for $P_5$-free graphs. The $P_6$-free case was the last open question for the complexity of ED on $F$-free graphs. Recently, Lokshtanov, Pilipczuk and van Leeuwen showed that weighted ED is solvable in polynomial time for $P_6$-free graphs, based on their sub-exponential algorithm for the Maximum Weight Independent Set problem for $P_6$-free graphs. Independently, at the same time, Mosca found a polynomial time algorithm for weighted ED on $P_6$-free graphs using a direct approach. In this paper, we describe the details of this approach which is simpler and much faster, namely its time bound is ${\cal O}(n^6 m)$

An O∗(1.1939n)O^*(1.1939^n) time algorithm for minimum weighted dominating induced matching

Say that an edge of a graph $G$ dominates itself and every other edge adjacent to it. An edge dominating set of a graph $G=(V,E)$ is a subset of edges $E' \subseteq E$ which dominates all edges of $G$. In particular, if every edge of $G$ is dominated by exactly one edge of $E'$ then $E'$ is a dominating induced matching. It is known that not every graph admits a dominating induced matching, while the problem to decide if it does admit it is NP-complete. In this paper we consider the problems of finding a minimum weighted dominating induced matching, if any, and counting the number of dominating induced matchings of a graph with weighted edges. We describe an exact algorithm for general graphs that runs in $O^*(1.1939^n)$ time and polynomial (linear) space. This improves over any existing exact algorithm for the problems in consideration.Comment: 17 page

Efficient Domination for Some Subclasses of P6P_6-Free Graphs in Polynomial Time

Let $G$ be a finite undirected graph. A vertex {\em dominates} itself and all its neighbors in $G$. A vertex set $D$ is an {\em efficient dominating set} (\emph{e.d.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.\ in $G$, is known to be \NP-complete even for very restricted graph classes such as $P_7$-free chordal graphs. The ED problem on a graph $G$ can be reduced to the Maximum Weight Independent Set (MWIS) problem on the square of $G$. The complexity of the ED problem is an open question for $P_6$-free graphs and was open even for the subclass of $P_6$-free chordal graphs. In this paper, we show that squares of $P_6$-free chordal graphs that have an e.d. are chordal; this even holds for the larger class of ($P_6$, house, hole, domino)-free graphs. This implies that ED/WeightedED is solvable in polynomial time for ($P_6$, house, hole, domino)-free graphs; in particular, for $P_6$-free chordal graphs. Moreover, based on our result that squares of $P_6$-free graphs that have an e.d. are hole-free and some properties concerning odd antiholes, we show that squares of ($P_6$, house)-free graphs (($P_6$, bull)-free graphs, respectively) that have an e.d. are perfect. This implies that ED/WeightedED is solvable in polynomial time for ($P_6$, house)-free graphs and for ($P_6$, bull)-free graphs (the time bound for ($P_6$, house, hole, domino)-free graphs is better than that for ($P_6$, house)-free graphs). The complexity of the ED problem for $P_6$-free graphs remains an open question

Dominating Induced Matchings for P8P_8-free Graphs in Polynomial Time

Let $G=(V,E)$ be a finite undirected graph. An edge set $E' \subseteq E$ is a dominating induced matching (d.i.m.) in $G$ if every edge in $E$ is intersected by exactly one edge of $E'$. The Dominating Induced Matching (DIM) problem asks for the existence of a d.i.m. 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 and is solvable in linear time for $P_7$-free graphs. However, its complexity was open for $P_k$-free graphs for any $k \ge 8$; $P_k$ denotes the chordless path with $k$ vertices and $k-1$ edges. We show in this paper that the weighted DIM problem is solvable in polynomial time for $P_8$-free graphs

Algorithmic problems in right-angled Artin groups: complexity and applications

In this paper we consider several classical and novel algorithmic problems for right-angled Artin groups, some of which are closely related to graph theoretic problems, and study their computational complexity. We study these problems with a view towards applications to cryptography.Comment: 16 page