On Locating-Chromatic Number for Graphs with Dominant Vertices

Let c be a k-coloring of a (not necessary connected) graph H. Let Pi = {C-1, C-2, ..., C-k} be the partition of V(H) induced by c, where C-i is partition class receiving color i. The color code c(Pi) (v) of a vertex v epsilon H is the ordered k-tuple (d(v, C-1), d(v, C-2), ..., d(v, C-k)), where d(v, C-i) = min{d(v, x)| x epsilon C-i} for all i epsilon [1, k]. If all vertices of H have distinct color codes, then c is called a locating k-coloring of H. The locating-chromatic number of H, denoted by chi(L)' (H), is the smallest k such that H admits a locatingcoloring with k colors. If there is no integer k satisfying the above conditions, then we say that chi(L)' (H) = infinity. Note that if H is a connected graph, then chi(L)' (H) = chi(L) (H). In this paper, we provide upper bounds for the locating-chromatic numbers of connected graphs obtained from disconnected graphs where each component contains a single dominant vertex. (C) 2015 The Authors. Published by Elsevier B.V.