On restrained coalitions in graphs: bounds and exact values

Paper

Abstract

A subset

is a dominating set of a graph

with vertex set

if every vertex

is adjacent to a vertex in

. Two subsets of

form a coalition if neither of them is a dominating set, but their union is a dominating set. A coalition partition of

is its vertex partition

such that every non-dominating set of

is a member of some coalition, and every dominating set is a single-vertex set in

. The coalition number

of a graph

is the maximum cardinality of its coalition partitions. A subset

is a restrained dominating set if

is a dominating set and any vertex of

has at least one neighbor in

. Restrained dominating coalition, restrained dominating partition and restrained coalition number

are defined by the same way. In this paper, we prove that

for an arbitrary graph

. In addition, the restrained coalition numbers of cycles and trees are determined.