Abstract
Graph Theory A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X-v)∪u is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. A graph G is p-stable if the largest arbitrary subset of edges whose removal from G does not increase the secure domination number of the resulting graph, has cardinality p. In this paper we study the problem of computing p-stable graphs for all admissible values of p and determine the exact values of p for which members of various infinite classes of graphs are p-stable. We also consider the problem of determining analytically the largest value ωn of p for which a graph of order n can be p-stable. We conjecture that ωn=n-2 and motivate this conjecture.
Highlights
Let G = (V, E) be a simple graph of order n
A dominating set X ⊆ V of G is secure dominating if, for each vertex u ∈ V − X, there exists a vertex v ∈ N (u) ∩ X such that the swap set (X − {v}) ∪ {u} is again a dominating set of G, where N (u) denotes the open neighbourhood of u
A vertex u ∈ V − X is an external private neighbour of a vertex v ∈ X with respect to some set X if N (u) ∩ X = {v} and a vertex w ∈ V is a universal vertex of G if it is adjacent all the vertices in V − {w}
Summary
In this paper we focus our attention on the dual problem of determining graphs G with the property that the largest number of arbitrary edges whose removal from G necessarily does not increase the secure domination number of the resulting graph, is a pre-specified value p This dual problem is relevant in the same generic application as described above, but where threshold information is sought with respect to the largest number of arbitrary edge failures which does not result in a requirement of additional guards to dominate the location complex securely.
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