Perfect edge domination: hard and solvable cases

Abstract

Let G be an undirected graph. An edge of G dominates itself and all edges adjacent to it. A subset E ′ of edges of G is an edge dominating set of G, if every edge of the graph is dominated by some edge of E ′ . We say that E ′ is a perfect edge dominating set of G, if every edge not in E ′ is dominated by exactly one edge of E ′ . The perfect edge dominating problem is to determine a least cardinality perfect edge dominating set of G. For this problem, we describe two NP-completeness proofs, for the classes of claw-free graphs of degree at most 3, and for bounded degree graphs, of maximum degree at most d ≥ 3 and large girth. In contrast, we prove that the problem admits an O(n) time solution, for cubic claw-free graphs. In addition, we prove a complexity dichotomy theorem for the perfect edge domination problem, based on the results described in the paper. Finally, we describe a linear time algorithm for finding a minimum weight perfect edge dominating set of a P 5 -free graph. The algorithm is robust, in the sense that, given an arbitrary graph G, either it computes a minimum weight perfect edge dominating set of G, or it exhibits an induced subgraph of G, isomorphic to a P 5 .

Some New Results on Efficient and Perfect Edge Domination of Graphs

“…It includes [2,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Papers on perfect edge domination are less frequent, The reader can refer to [1,20,21,22]. In the latter paper, the authors describe ILP formulations for the PED problem, together with some experimental results.…”

“…In [1], the authors proved that neighborhood-star-free graphs do not have any proper perfect dominating sets. The only possible PEDs of these graphs are (i) the trivial PED, and (ii) EEDs.…”

Modelling and solving the perfect edge domination problem

“…PEDP is known to be NP-hard for some specific graph classes. Among them, bipartite graphs [21] and K 3 -free 3-regular graphs [17], where K 3 stands for a complete graph on three vertices. We have thus devised generation schemes that produce particular graphs belonging to these two classes.…”

“…It is NP-complete to determine if a given graph contains a PEDS of a given size [21] and that result also holds true for claw-free graphs of degree at most 3, bipartite graphs [21], k-regular graphs with k ≥ 3, and bounded-degree graphs of large girth or bounded-degree Ffree graphs, except when F is a set of disjoint paths (see [17], for all of these particular cases). Polynomial time algorithms are known for chordal graphs [21], circular-arc graphs [19], P 5 -free graphs [17], cubic claw-free graphs and bounded-degree F -free graphs [17], where F is a set of disjoint paths. No exact algorithm, combinatorial or IP based, appears to exist for PEDP.…”