### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 574-581 |

Number of pages | 8 |

Journal | IEICE Transactions on Information and Systems |

Volume | E101D |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 1 2018 |

### All Science Journal Classification (ASJC) codes

- Software
- Hardware and Architecture
- Computer Vision and Pattern Recognition
- Electrical and Electronic Engineering
- Artificial Intelligence

### Cite this

*IEICE Transactions on Information and Systems*,

*E101D*(3), 574-581. https://doi.org/10.1587/transinf.2017FCP0007

**Complexity of the minimum single dominating cycle problem for graph classes.** / Eto, Hiroshi; Kawahara, Hiroyuki; Miyano, Eiji; Nonoue, Natsuki.

Research output: Contribution to journal › Article

*IEICE Transactions on Information and Systems*, vol. E101D, no. 3, pp. 574-581. https://doi.org/10.1587/transinf.2017FCP0007

}

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

PY - 2018/3/1

Y1 - 2018/3/1

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 -