Roman domination with respect to nondegenerate graph properties: vertex and edge removal

For a graph property P and a graph G, a subset S of the vertices of G is a P-set if the subgraph induced by S has the property P. A P-Roman dominating function on a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2 and the set of all vertices with label 1 or 2 is a P-set. The P-Roman domination number γPR(G) of G is the minimum of Σv∈V (G)f(v) over such functions. In this paper we present results on changing and unchanging of γPR(G) when a graph is modified by deleting an edge or a vertex. Some known results for the ordinary Roman domination number are extended and generalized to γPR(G). The P-Roman bondage number bPR(G) is the cardinality of a smallest set of edges whose removal from G results in a graph with P-Roman domination number not equal to γPR(G). We obtain upper bounds in terms of (a) edge degree and maximum degree, (b) average degree and maximum degree, (c) orientable/non orientable genus and maximum degree, and (d) Euler characteristic, girth and maximum degree, for the P-Roman bondage number of a graph on topological surfaces. We also prove that for any graph G, which admits a 2-cell embedding on a surface with non-negative Euler characteristic, either bPR(G) ≤ 15 or 15 < bPR(G) ≤ Δ(G)− 3.