Minimum Degree and Dominating Paths
Ralph J. Faudree, Ronald J. Gould, Michael S. Jacobson, Douglas B. West · Journal of Graph Theory · 2016
Abstract 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 , 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 . For , 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 , an s‐vertex dominating path is guaranteed by when n is sufficiently large, but (where ) 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.