Efficient Domination Research Articles

On efficient domination for some classes of [formula omitted]-free bipartite graphs

A vertex set D in a finite undirected graph G is an efficient dominating set (e.d.s. for short) of G if every vertex of G is dominated by exactly one vertex of D. The 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 2P3-free chordal graphs while it is solvable in polynomial time for P6-free graphs. Here we focus on bipartite graphs: We show that (weighted) ED can be solved in polynomial time for H-free bipartite graphs when H is P7 or ℓP4 for fixed ℓ, and similarly for P9-free bipartite graphs with vertex degree at most 3, and when H is S2,2,4. Moreover, we show that ED is NP-complete for bipartite graphs with diameter at most 6.

On efficient domination for some classes of [formula omitted]-free chordal graphs

A vertex set D in a finite undirected graph G is an efficient dominating set (e.d.s for short) of G if every vertex of G is dominated by exactly one vertex of D. The 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 graph classes such as for 2P3-free chordal graphs while it is solvable in polynomial time for P6-free chordal graphs (and even for P6-free graphs). A standard reduction from the NP-complete Exact Cover problem shows that ED is NP-complete for a very special subclass of chordal graphs generalizing split graphs. The reduction implies that ED is NP-complete e.g. for double-gem-free chordal graphs while it is solvable in linear time for gem-free chordal graphs (by various reasons such as bounded clique-width, distance-hereditary graphs, chordal square etc.), and ED is NP-complete for butterfly-free chordal graphs while it is solvable in linear time for 2P2-free graphs.We show that (weighted) ED can be solved in polynomial time for H-free chordal graphs when H is net, extended gem, or S1,2,3.

A dichotomy for weighted efficient dominating sets with bounded degree vertices

In a finite undirected graph G, a vertex v dominates itself and its neighbors. A vertex set D is an efficient dominating set (e.d.s. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in G, is well known to be NP-complete for graphs of maximum degree 3. We say that an e.d.s. D of G is k-bounded (k-b.e.d.s. for short) if the degree of every vertex in D is at most k in G. The task of the k-Bounded Weighted Efficient Domination (k-BWED) problem is to determine whether a given vertex-weighted graph G admits a k-b.e.d.s., and if so, to compute one of minimum weight. It easily follows from the NP-completeness of ED for graphs of maximum degree 3 that the k-BWED problem is NP-complete for every k≥3, and clearly, the k-BWED problem is solvable in linear time for k≤1. In this note, we show that the 2-BWED problem is solvable in time O(|V(G)|(|V(G)|+|E(G)|)), thus obtaining a dichotomy of the complexity status of k-BWED over all k≥0.

Polynomial-time algorithm for weighted efficient domination problem on diameter three planar graphs

A set D of vertices in a graph G is an efficient dominating set (e.d.s. for short) of G if D is an independent set and every vertex not in D is adjacent to exactly one vertex in D. The efficient domination (ED) problem asks for the existence of an e.d.s. in G. The minimum weighted efficient domination problem (MIN-WED for short) is the problem of finding an e.d.s. of minimum weight in a given vertex-weighted graph. Brandstädt, Fičur, Leitert and Milanič (2015) [3] stated the running times of the fastest known polynomial-time algorithms for the MIN-WED problem on some graphs classes by using a Hasse diagram.In this paper, we update this Hasse diagram by showing that, while for every integer d such that d=3k or d=3k+2, where k≥1, the ED problem remains NP-complete for graphs of diameter d, the weighted version of the problem is solvable in time O(|V(G)|+|E(G)|) in the class of diameter three bipartite graphs and in time O(|V(G)|5) in the class of diameter three planar graphs.

Weighted efficient domination for some classes of [formula omitted]-free and of [formula omitted]-free graphs

A vertex set D in a finite undirected graph G is an efficient dominating set (e.d.s. for short) of G if every vertex of G is dominated by exactly one vertex of D. The 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 graph classes such as for claw-free graphs, for 2P3-free chordal graphs (and thus, for P7-free graphs), and for bipartite graphs. For the complexity of ED and its weighted version WED, a dichotomy for H-free graphs was reached: A graph H is called a linear forest if H is acyclic and claw-free, that is, if all its components are paths. Thus, the ED problem remains NP-complete for H-free graphs whenever H is not a linear forest. For every linear forest H, WED is either solvable in polynomial time or NP-complete for H-free graphs; the final result showed that WED is solvable in polynomial time for P6-free graphs.For (H1,H2)-free graphs, however, we are still far away from a dichotomy result. The main topics of this paper are: (1) to improve the time bounds and simplify the proofs (based on modular decomposition) for polynomial time cases of WED for some H-free graph classes; (2) to investigate the complexity of WED for some cases of (H1,H2)-free graphs such that WED is NP-complete for Hi-free graphs for at least one of i∈{1,2}. Since it is well known that WED is solvable in polynomial time for graph classes of bounded clique-width, we consider only classes of (H1,H2)-free graphs with unbounded clique-width.

On Efficient Domination for Some Classes of H-Free Chordal Graphs

A vertex set D in a finite undirected graph G is an efficient dominating set (e.d.s. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem asks for the existence of an e.d.s. in G, and the Weighted Efficient Domination (WED) problem asks for such an e.d.s. of minimum weight. While, based on [A. Brandstädt and V. Giakoumakis, Weighted Efficient Domination for (P5 + kP2)-Free Graphs in Polynomial Time, CoRR arXiv:1407.4593, 2014; A. Brandstädt, M. Milanič, and R. Nevries, New polynomial cases of the weighted efficient domination problem, extended abstract in: Conference Proceedings of MFCS 2013, LNCS 8087, 2013, 195–206; Full version: CoRR arXiv:1304.6255, 2013], for the complexity of WED on H-free graphs, a dichotomy was reached—see [A. Brandstädt and R. Mosca, Weighted efficient domination for P5-free and P6-free graphs, extended abstract in: Proceedings of WG 2016, P. Heggernes, ed., LNCS 9941, pp. 38–49, 2016; Journal version: SIAM J. Discrete Math. 30, 4 (2016) 2288–2303; D. Lokshtanov, M. Pilipczuk, and E.J. van Leeuwen, Independence and Efficient Domination on P6-Free Graphs, Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA) 2016, 1784–1803.]—it is still an open problem for many classes of H-free chordal graphs; a standard reduction from the NP-complete Exact Cover problem shows that WED is NP-complete for a very special subclass of chordal graphs (slightly generalizing split graphs—see the forbidden induced subgraphs such as 2P3 and seven other examples in the figure). The main results in this paper are polynomial-time solutions of WED for H-free chordal graphs, when H is co-P, net, S1,2,2, and S1,2,3.

Efficient domination for classes of [formula omitted]-free graphs

In a finite undirected graph G, a vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The 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 P7-free chordal graphs; its complexity was an open question for P6-free graphs and was open even for the subclass of P6-free chordal graphs. Recently, Lokshtanov et al., and independently, Brandstädt and Mosca showed that ED is solvable in polynomial time for P6-free graphs.The ED problem on a graph G can be reduced to the Maximum Weight Independent Set problem on the square of G. In this paper, we show that squares of P6-free chordal graphs that have an e.d. are chordal; this even holds for the larger class of (P6, house, hole, domino)-free graphs. Thus, ED/WeightedED is solvable in polynomial time for (P6, house, hole, domino)-free graphs, and in particular, for P6-free chordal graphs. Also, the time bound achieved for ED on this class is much better than in the P6-free case.Moreover, we show that squares of P6-free graphs that have an e.d. are hole-free. Based on this result, we show that ED is solvable in polynomial time for (P6, net)-free graphs; again, the time bound for ED on this class is much better than in the P6-free case.

Weighted Efficient Domination for $P_5$-Free and $P_6$-Free Graphs

In a finite undirected graph $G=(V,E)$, a vertex $v \in V$ dominates itself and its neighbors in $G$. A vertex set $D \subseteq V$ is an efficient dominating set (e.d.s. for short) of $G$ if every $v \in V$ is dominated in $G$ by exactly one vertex of $D$. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in $G$, is known to be NP-complete for $P_7$-free graphs and 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 quasi-polynomial algorithm for the Maximum Weight Independent Set problem for $P_6$-free graphs. Independently, by a direct approach which is simpler and faster, we found an ${\cal O}(n^5 m)$ time solution for weighted ED on $P_6$-free graphs. Moreover, we show that weighted ED is solvable in linear time for $P_5$-free graphs which so...