
For FullText PDF, please login, if you are a member of IEICE,
or go to Pay Per View on menu list, if you are a nonmember of IEICE.

Complexity of the Minimum Single Dominating Cycle Problem for Graph Classes
Hiroshi ETO Hiroyuki KAWAHARA Eiji MIYANO Natsuki NONOUE
Publication
IEICE TRANSACTIONS on Information and Systems
Vol.E101D
No.3
pp.574581 Publication Date: 2018/03/01 Publicized: 2017/12/19 Online ISSN: 17451361
DOI: 10.1587/transinf.2017FCP0007 Type of Manuscript: Special Section PAPER (Special Section on Foundations of Computer Science — Frontiers of Theoretical Computer Science —) Category: Keyword: minimum single dominating problem, graph classes, (in)tractability, (in)approximability,
Full Text: PDF(456.6KB)>>
Summary:
In this paper, we study a variant of the MINIMUM DOMINATING SET problem. Given an unweighted undirected graph G=(V,E) of n=V vertices, the goal of the MINIMUM SINGLE DOMINATING CYCLE problem (MinSDC) is to find a single shortest cycle which dominates all vertices, i.e., a cycle C such that for the set V(C) of vertices in C and the set N(V(C)) of neighbor vertices of C, V(G)=V(C)∪N(V(C)) and V(C) is minimum over all dominating cycles in G [6], [17], [24]. In this paper we consider the (in)approximability of MinSDC if input graphs are restricted to some special classes of graphs. We first show that MinSDC is still NPhard to approximate even when restricted to planar, bipartite, chordal, or rregular (r≥3). Then, we show the (lnn+1)approximability and the (1ε)lnninapproximability of MinSDC on split graphs under P≠NP. Furthermore, we explicitly design a lineartime algorithm to solve MinSDC for graphs with bounded treewidth and estimate the hidden constant factor of its running timebound.

