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

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

Connected Vertex Cover for (sP1+P5)(sP_1+P_5)-Free Graphs

The Connected Vertex Cover problem is to decide if a graph G has a vertex cover of size at most $k$ that induces a connected subgraph of $G$. This is a well-studied problem, known to be NP-complete for restricted graph classes, and, in particular, for $H$-free graphs if $H$ is not a linear forest (a graph is $H$-free if it does not contain $H$ as an induced subgraph). It is easy to see that Connected Vertex Cover is polynomial-time solvable for $P_4$-free graphs. We continue the search for tractable graph classes: we prove that it is also polynomial-time solvable for $(sP_1+P_5)$-free graphs for every integer $s\geq 0$

Independent Sets in Classes Related to Chair/Fork-free Graphs

The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. MWIS is known to be $NP$-complete in general, even under various restrictions. Let $S_{i,j,k}$ be the graph consisting of three induced paths of lengths $i, j, k$ with a common initial vertex. The complexity of the MWIS problem for $S_{1, 2, 2}$-free graphs, and for $S_{1, 1, 3}$-free graphs are open. In this paper, we show that the MWIS problem can solved in polynomial time for ($S_{1, 2, 2}$, $S_{1, 1, 3}$, co-chair)-free graphs, by analyzing the structure of the subclasses of this class of graphs. This extends some known results in the literature.Comment: arXiv admin note: text overlap with arXiv:1504.0540

The maximum weight stable set problem in (P_6,\mbox{bull})-free graphs

We present a polynomial-time algorithm that finds a maximum weight stable set in a graph that does not contain as an induced subgraph an induced path on six vertices or a bull (the graph with vertices $a, b, c, d, e$ and edges $ab, bc, cd, be, ce$)

Weighted Independent Sets in a Subclass of P6P_6-free Graphs

The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. The complexity of the MWIS problem for $P_6$-free graphs is unknown. In this note, we show that the MWIS problem can be solved in time $O(n^3m)$ for ($P_6$, banner)-free graphs by analyzing the structure of subclasses of these class of graphs. This extends the existing results for ($P_5$, banner)-free graphs, and ($P_6$, $C_4$)-free graphs. Here, $P_t$ denotes the chordless path on $t$ vertices, and a banner is the graph obtained from a chordless cycle on four vertices by adding a vertex that has exactly one neighbor on the cycle.Comment: arXiv admin note: text overlap with arXiv:1503.0602

Reed's conjecture on some special classes of graphs

Reed conjectured that for any graph $G$, $\chi(G) \leq \lceil \frac{\omega(G)+\Delta(G)+1}{2}\rceil$, where $\chi(G)$, $\omega(G)$, and $\Delta(G)$ respectively denote the chromatic number, the clique number and the maximum degree of $G$. In this paper, we verify this conjecture for some special classes of graphs, in particular for subclasses of $P_5$-free graphs or $Chair$-free graphs.Comment: Submitted to Discrete Mathematic

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

Hereditary Graph Classes: When the Complexities of Colouring and Clique Cover Coincide

A graph is $(H_1,H_2)$-free for a pair of graphs $H_1,H_2$ if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. In 2001, Kr\'al', Kratochv\'{\i}l, Tuza, and Woeginger initiated a study into the complexity of Colouring for $(H_1,H_2)$-free graphs. Since then, others have tried to complete their study, but many cases remain open. We focus on those $(H_1,H_2)$-free graphs where $H_2$ is $\overline{H_1}$, the complement of $H_1$. As these classes are closed under complementation, the computational complexities of Colouring and Clique Cover coincide. By combining new and known results, we are able to classify the complexity of Colouring and Clique Cover for $(H,\overline{H})$-free graphs for all cases except when $H=sP_1+ P_3$ for $s\geq 3$ or $H=sP_1+P_4$ for $s\geq 2$. We also classify the complexity of Colouring on graph classes characterized by forbidding a finite number of self-complementary induced subgraphs, and we initiate a study of $k$-Colouring for $(P_r,\overline{P_r})$-free graphs.Comment: 19 Pages, 5 Figure