Abstract
A 2-rainbow domination function of a graph G=(V,E) is a function f mapping each vertex v to a subset of {1,2} such that ⋃u∈N(v)f(u)={1,2} when f(v)=∅, where N(v) is the open neighborhood of v. The weight of f is denoted by wf(G)=∑v∈V|f(v)|. The 2-rainbow domination number, denoted by γr2(G), is the smallest wf(G) among all 2-rainbow domination functions f of G. The 2-rainbow bondage number, denoted by br2(G), is the minimum cardinality among all sets E′⊆E such that γr2(G-E′)>γr2(G), where G-E′ denotes the resulting graph after all edges in E′ are removed from G. In this paper, we show that br2(P(n,2))=2 when n⩾15 and n≡3,9(mod10), and br2(P(n,2))⩽4 when n≢3,9(mod10).
Highlights
Let G = (V, E) be an undirected graph, where V(G) and E(G) are vertex and edge sets of G, respectively
Kuo-Hua Wu received his PhD degree in information management from National Taiwan University of Science and Technology (NTUST). He is working on an IoT Security project in Institute for Information Industry in Taipei
We investigate the 2-rainbow bondage problem on generalized Petersen graphs
Summary
The k-rainbow bondage problem is to investigate the problem of removing the minimum number of edges from a graph such that the k-rainbow domination number is increased. We investigate the 2-rainbow bondage problem on generalized Petersen graphs. There are a lot of papers investigating the bondage problem which is the problem of removing the minimum number of edges from a graph such that the domination number is increased. The degree of a vertex v in a graph G is the number of edges incident with v and is denoted by deg(v). In Bresă r, Henning, and Rall (2008) defined the k-rainbow domination number as follows: Definition 1.1 A k-rainbow domination function (k-RDF for short) of G is a function f mapping each vertex v to a subset of
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