Detour Domination in Graphs

Abstract

For distinct vertices \(u\) and \(v\) of a nontrivial connected graph \(G\), the detour distance \(D(u,v)\) between \(u\) and \(v\) is the length of a longest \(u-v\) path in \(G\). For a vertex \(v \in V(G)\), define \(D^-(v) = \min\{D(u,v) : u \in V(G) – \{v\}\}\). A vertex \(u (\neq v)\) is called a detour neighbor of \(v\) if \(D(u,v) = D^-(v)\). A vertex \(v\) is said to detour dominate a vertex \(u\) if \(u = v\) or \(u\) is a detour neighbor of \(v\). A set \(S\) of vertices of \(G\) is called a detour dominating set if every vertex of \(G\) is detour dominated by some vertex in \(S\). A detour dominating set of \(G\) of minimum cardinality is a minimum detour dominating set and this cardinality is the detour domination number \(\gamma_D(G)\). We show that if \(G\) is a connected graph of order \(n \geq 3\), then \(\gamma_D(G) \leq n-2\). Moreover, for every pair \(k,n\) of integers with \(1 \leq k \leq n-2\), there exists a connected graph \(G\) of order \(n\) such that \(\gamma_D(G) = k\). It is also shown that for each pair \(a,b\) of positive integers, there is a connected graph \(G\) with domination number \(\gamma(G) = a\) and \(\gamma_D(G) = b\).