Bounds on the signed distance-�����-domination number of graphs

Abstract

Let 􀜩 􀵌 􁈺􀜸, 􀜧􁈻 be a graph with vertex set 􀜸 􀵌 􀜸􁈺􀜩􁈻 of order 􀝊 and edge set 􀜧 􀵌 􀜧􁈺􀜩􁈻. A 􀝇 -dominating set of 􀜩 is a subset 􀜵 􀙃 􀜸 such that each vertex in 􀜸 􀜵 has at least 􀝇 neighbors in 􀜵. If 􀝒 is a vertex of a graph 􀜩, the open 􀝇-neighborhood of 􀝒, denoted by 􀜰􀯞􁈺􀝒􁈻, is the set 􀜰􀯞􁈺􀝒􁈻 􀵌 􁈼􀝑 􀗐 􀜸 􀗷 􀝑 􀵍 􀝒 􀜽􀝊􀝀 􀝀􁈺􀝑, 􀝒􁈻 􀵑 􀝇 􁈽. 􀜰􀯞􁈾􀝒􁈿 􀵌 􀜰􀯞􁈺􀝒􁈻 􀗫 􁈼􀝒􁈽 is the closed 􀝇-neighborhood of 􀝒. A function 􀝂 􀗷 􀜸 􀸷 􁈼􀵆1, 1􁈽 is a signed distance-􀝇 dominating function of 􀜩, if for every vertex 􀝒 􀗐 􀜸, 􀝂􁈺􀜰􀯞􁈾􀝒􁈿􁈻 􀵌 Σ􀯨 􀗐 􀯇􀳖􁈾􀯩􁈿 􀝂􁈺􀝑􁈻 􀵒 1. The signed distance-􀝇-domination number, denoted by 􀟛􀯞,􀯦􁈺􀜩􁈻, is the minimum weight of a signed distance-􀝇-dominating function of 􀜩. In this paper, we give lower and upper bounds on 􀟛􀯞,􀯦 of graphs. Also, we determine the signed distance-􀝇-domination number of graph 􀟛􀯞,􀯦􁈺􀜩 􀗩 􀜪􁈻 (the graph obtained from the disjoint union 􀜩 􀵅 􀜪 by adding the edges 􁈼􀝔􀝕 􀗷 􀝔 􀗐 􀜸􁈺􀜩􁈻, 􀝕 􀗐 􀜸􁈺􀜪􁈻􁈽) when 􀝇 􀵒 2.