### Abstract

Let G=(V,E) be a graph, and w:V→Q_{>0} be a positive weight function on the vertices of G. For every subset X of V, let w(X)=∑_{v∈G}w(v). A non-empty subset S⊂V(G) is a weighted safe set if, for every component C of the subgraph induced by S and every component D of G\S, we have w(C)≥w(D) whenever there is an edge between C and D. In this paper we show that the problem of computing the minimum weight of a safe set is NP-hard for trees, even if the underlining tree is restricted to be a star, but it is polynomially solvable for paths. Then we define the concept of a parameterized infinite family of “proper central subgraphs” on trees, whose polar ends are the minimum-weight connected safe sets and the centroids. We show that each of these central subgraphs includes a centroid. We also give a linear-time algorithm to find all of these subgraphs on unweighted trees.

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

Pages (from-to) | 79-84 |

Number of pages | 6 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 54 |

DOIs | |

Publication status | Published - Oct 1 2016 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Electronic Notes in Discrete Mathematics*,

*54*, 79-84. https://doi.org/10.1016/j.endm.2016.09.015

**Network Majority on Tree Topological Network.** / Bapat, Ravindra B.; Fujita, Shinya; Legay, Sylvain; Manoussakis, Yannis; Matsui, Yasuko; Sakuma, Tadashi; Tuza, Z.

Research output: Contribution to journal › Article

*Electronic Notes in Discrete Mathematics*, vol. 54, pp. 79-84. https://doi.org/10.1016/j.endm.2016.09.015

}

TY - JOUR

T1 - Network Majority on Tree Topological Network

AU - Bapat, Ravindra B.

AU - Fujita, Shinya

AU - Legay, Sylvain

AU - Manoussakis, Yannis

AU - Matsui, Yasuko

AU - Sakuma, Tadashi

AU - Tuza, Z.

PY - 2016/10/1

Y1 - 2016/10/1

N2 - Let G=(V,E) be a graph, and w:V→Q>0 be a positive weight function on the vertices of G. For every subset X of V, let w(X)=∑v∈Gw(v). A non-empty subset S⊂V(G) is a weighted safe set if, for every component C of the subgraph induced by S and every component D of G\S, we have w(C)≥w(D) whenever there is an edge between C and D. In this paper we show that the problem of computing the minimum weight of a safe set is NP-hard for trees, even if the underlining tree is restricted to be a star, but it is polynomially solvable for paths. Then we define the concept of a parameterized infinite family of “proper central subgraphs” on trees, whose polar ends are the minimum-weight connected safe sets and the centroids. We show that each of these central subgraphs includes a centroid. We also give a linear-time algorithm to find all of these subgraphs on unweighted trees.

AB - Let G=(V,E) be a graph, and w:V→Q>0 be a positive weight function on the vertices of G. For every subset X of V, let w(X)=∑v∈Gw(v). A non-empty subset S⊂V(G) is a weighted safe set if, for every component C of the subgraph induced by S and every component D of G\S, we have w(C)≥w(D) whenever there is an edge between C and D. In this paper we show that the problem of computing the minimum weight of a safe set is NP-hard for trees, even if the underlining tree is restricted to be a star, but it is polynomially solvable for paths. Then we define the concept of a parameterized infinite family of “proper central subgraphs” on trees, whose polar ends are the minimum-weight connected safe sets and the centroids. We show that each of these central subgraphs includes a centroid. We also give a linear-time algorithm to find all of these subgraphs on unweighted trees.

UR - http://www.scopus.com/inward/record.url?scp=84992536188&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84992536188&partnerID=8YFLogxK

U2 - 10.1016/j.endm.2016.09.015

DO - 10.1016/j.endm.2016.09.015

M3 - Article

AN - SCOPUS:84992536188

VL - 54

SP - 79

EP - 84

JO - Electronic Notes in Discrete Mathematics

JF - Electronic Notes in Discrete Mathematics

SN - 1571-0653

ER -