Edge Roman Domination on Graphs

Abstract

An edge Roman dominating function of a graph G is a function $f:E(G) \rightarrow \{0,1,2\}$ 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 $\gamma '_R(G)$, is the minimum weight $w(f) = \sum _{e\in E(G)} f(e)$ 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 $\Delta$ on n vertices, then $\gamma _R'(G) \le \lceil \frac{\Delta }{\Delta +1} n \rceil$. While the counterexamples having the edge Roman domination numbers $\frac{2\Delta -2}{2\Delta -1} n$, we prove that $\frac{2\Delta -2}{2\Delta -1} n + \frac{2}{2\Delta -1}$ 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 $\frac{6}{7}n$, 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 $K_{2,3}$ as a subdivision, which generalizes a result of Akbari and Qajar on outerplanar graphs.