Minimum maximal graphs with forbidden subgraphs
A vertex k-ranking of a simple graph is a coloring of its vertices with k colors in such a way that each path connecting two vertices of the same color contains a vertex with a bigger color. Consider the minimum vertex ranking spanning tree (MVRST) problem where the goal is to find a spanning tree of a given graph G which has a vertex ranking using the minimal number of colors over vertex rankings of all spanning trees of G. K. Miyata et al. proved in [NP-hardness proof and an approximation algorithm...
A three-valued function defined on the vertices of a graph is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. That is, for every , , where consists of every vertex adjacent to . The weight of an MTDF is , over all vertices . The minus total domination number of a graph , denoted , equals the minimum weight of an MTDF of . In this paper, we discuss some properties of minus total domination on a graph and obtain...
A celebrated result of Bringmann and Ono shows that the combinatorial rank generating function exhibits automorphic properties after being completed by the addition of a non-holomorphic integral. Since then, automorphic properties of various related combinatorial families have been studied. Here, extending work of Andrews and Bringmann, we study general infinite families of combinatorial q-series pertaining to k-marked Durfee symbols, in which we allow additional singularities. We show that these...