Minimum Degree and Dominating Paths

A dominating path in a graph is a path P such that every vertex outside P has a neighbor on P. A result of Broersma from 1988 implies that if G is an n-vertex k-connected graph and δ(G) > n−k k+2 − 1, then G contains a dominating path. We prove the following results. The lengths of dominating paths include all values from the shortest up to at least min{n − 1,2δ(G)}. For δ(G) > an, where a is a constant greater than 1/3, the minimum length of a dominating path is at most logarithmic in n when n is sufficiently large (the base of the logarithm depends on a). The preceding results are sharp. For constant s and c′ < 1, an s-vertex dominating path is guaranteed by δ(G) ≥ n − 1 − c′n1−1/s when n is sufficiently large, but δ(G) ≥ n − c(s ln n)1/sn1−1/s (where c > 1) does not even guarantee a dominating set of size s. We also obtain minimum-degree conditions for the existence of a spanning tree obtained from a dominating path by giving the same number of leaf neighbors to each vertex. C © 2016 Wiley Periodicals, Inc. J. Graph Theory 84: 202–213, 2017