Exact Algorithms for Annotated Edge Dominating Set in Graphs with Degree Bounded by 3

Abstract

SUMMARYGiven a graph G = (V, E) together with a nonnegative integer requirement on vertices r : V → Z + , the annotated edge dominating set problem is to find a minimum set M ⊆ E such that, each edge in E − M is adjacent to some edge in M, and M contains at least r(v) edges incident on each vertex v ∈ V. The annotated edge dominating set problem is a natural extension of the classical edge dominating set problem, in which the requirement on vertices is zero. The edge dominating set problem is an important graph problem and has been extensively studied. It is well known that the problem is NP-hard, even when the graph is restricted to a planar or bipartite graph with maximum degree 3. In this paper, we show that the annotated edge dominating set problem in graphs with maximum degree 3 can be solved in O * (1.2721 n ) time and polynomial space, where n is the number of vertices in the graph. We also show that there is an O * (2.2306 k )-time polynomial-space algorithm to decide whether a graph with maximum degree 3 has an annotated edge dominating set of size k or not.

An Improved Algorithm for Parameterized Edge Dominating Set Problem

“…Case (a). The algorithm branches on v (or the admissible 4-cycle on it) in G[U ′ 2 ] with the recurrence (12) and exactly one bad component H is produced in one of the first and second branches: When H is a 2-path component, we have the recurrence (9). When H is a bi-claw or leg-triangle component, we have the recurrence (10).…”

A Refined Exact Algorithm for Edge Dominating Set

“…Given a graph G, a subset C of vertices of G is called a vertex cover, if any edge in G is incident on at least one vertex in C. A subset I of vertices of G is called an independent set, if there is no edge between any two vertices in I. EDS is an important problem studied from the view of enumeration vertex covers [3,11,4,14]. Note that the vertex set of an edge dominating set is a vertex cover.…”

“…By using the Measure and Conquer method, Rooij and Bodlaender [11] designed a simple O * (1.3323 n )-time algorithm, and further improved the running time bound to O * (1.3226 n ) by checking a number of local structures. When the graph is restricted to graphs of maximal degree three, the result can be improved to O * (1.2721 n ) [14]. There are also a numerous contributions to the parameterized algorithms for EDS with parameter k being the size of the edge dominating set [3,4,1,15,13].…”

An Exact Algorithm for Lowest Edge Dominating Set

“…Analyzing with the measure and conquer method, van Rooij and Bodlaender [10] designed an O * (1.3226 n )-time and polynomial-space algorithm and later Xiao and Nagamochi [14] presented an O * (1.3160 n )-time and polynomial-space algorithm, which currently attains the best time bound to Minimum Edge Dominating Set. For graphs of maximum degree 3, an O * (1.2721 n )-time and polynomial-space algorithm is designed by Xiao and Nagamochi [15].…”