Degree sum and vertex dominating paths

A vertex dominating path in a graph is a path P such that every vertex outside P has a neighbor on P. In 1988 H. Broersma [5] stated a result implying that every n‐vertex k‐connected graph G such that σ(k+2)(G)≥n−2k−1 contains a vertex dominating path. We provide a short, self‐contained proof of this result and further show that every n‐vertex k‐connected graph such that σ2(G)≥2nk+2+f(k) contains a vertex dominating path of length at most (20k)|T|, where T is a minimum dominating set of vertices. An immediate corollary of this result is that every such graph contains a vertex dominating path with length bounded above by a logarithmic function of the order of the graph. To derive this result, we prove that every n‐vertex k‐connected graph with σ2(G)≥2nk+2+f(k) contains a path of length at most 20k|T|, through any set of T vertices where |T|≤n/900k4.