TY - JOUR

T1 - Complexity of the minimum single dominating cycle problem for graph classes

AU - Eto, Hiroshi

AU - Kawahara, Hiroyuki

AU - Miyano, Eiji

AU - Nonoue, Natsuki

N1 - Funding Information:
This work is partially supported by JSPS KAKENHI JP15J05484, JP26330017 and JP17K00016. The authors would like to thank the anonymous reviewers for their detailed comments and suggestions that helped to improve the presentation of the paper.
Publisher Copyright:
© 2018 The Institute of Electronics, Information and Communication Engineers.

PY - 2018/3

Y1 - 2018/3

N2 - 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 NP-hard to approximate even when restricted to planar, bipartite, chordal, or r-regular (r ≥ 3). Then, we show the (ln n + 1)-approximability and the (1 - ϵ) ln n-inapproximability of MinSDC on split graphs under P ≠ NP. Furthermore, we explicitly design a linear-time algorithm to solve MinSDC for graphs with bounded treewidth and estimate the hidden constant factor of its running time-bound.

AB - 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 NP-hard to approximate even when restricted to planar, bipartite, chordal, or r-regular (r ≥ 3). Then, we show the (ln n + 1)-approximability and the (1 - ϵ) ln n-inapproximability of MinSDC on split graphs under P ≠ NP. Furthermore, we explicitly design a linear-time algorithm to solve MinSDC for graphs with bounded treewidth and estimate the hidden constant factor of its running time-bound.

UR - http://www.scopus.com/inward/record.url?scp=85042631622&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85042631622&partnerID=8YFLogxK

U2 - 10.1587/transinf.2017FCP0007

DO - 10.1587/transinf.2017FCP0007

M3 - Article

AN - SCOPUS:85042631622

VL - E101D

SP - 574

EP - 581

JO - IEICE Transactions on Information and Systems

JF - IEICE Transactions on Information and Systems

SN - 0916-8532

IS - 3

ER -