Edge Roman Domination on Graphs

Gerard J. Chang, Sheng Hua Chen, Chun-Hung Liu

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4 Scopus citations

Abstract

An edge Roman dominating function of a graph G is a function f: E(G) → { 0 , 1 , 2 } satisfying the condition that every edge e with f(e) = 0 is adjacent to some edge e with f(e) = 2. The edge Roman domination number of G, denoted by γR′(G), is the minimum weight w(f) = ∑ e E ( G )f(e) of an edge Roman dominating function f of G. This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad stating that if G is a graph of maximum degree Δ on n vertices, then γR′(G)≤⌈ΔΔ+1n⌉. While the counterexamples having the edge Roman domination numbers 2Δ-22Δ-1n, we prove that 2Δ-22Δ-1n+22Δ-1 is an upper bound for connected graphs. Furthermore, we provide an upper bound for the edge Roman domination number of k-degenerate graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic graphs. In addition, we prove that the edge Roman domination numbers of planar graphs on n vertices is at most 67n, which confirms a conjecture of Akbari and Qajar. We also show an upper bound for graphs of girth at least five that is 2-cell embeddable in surfaces of small genus. Finally, we prove an upper bound for graphs that do not contain K2 , 3 as a subdivision, which generalizes a result of Akbari and Qajar on outerplanar graphs.

Original languageEnglish (US)
Pages (from-to)1731-1747
Number of pages17
JournalGraphs and Combinatorics
Volume32
Issue number5
DOIs
StatePublished - Sep 1 2016

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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