Graph compositions. I: Basic enumeration.
Let G = (V,E) be a graph, and k ≥ 1 an integer. A subgraph D is said to be k-dominating in G if every vertex of G-D is at distance at most k from some vertex of D. For a given class of graphs, Domₖ is the set of those graphs G in which every connected induced subgraph H has some k-dominating induced subgraph D ∈ which is also connected. In our notation, Dom coincides with Dom₁. In this paper we prove that holds for = all connected graphs without induced (u ≥ 2). (In particular, ₂ = K₁ and...
PageRank is a ranking method that assigns scores to web pages using the limit distribution of a random walk on the web graph. A fibration of graphs is a morphism that is a local isomorphism of in-neighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. We show that a deep connection relates fibrations and Markov chains with restart, a particular kind of Markov chains that include the PageRank one as a special case. This fact provides constraints on the...
Let be a graph. A vertex subversion strategy of , say , is a set of vertices in whose closed neighborhood is removed from . The survival-subgraph is denoted by . The Neighbor-Integrity of , , is defined to be , where is any vertex subversion strategy of , and is the maximum order of the components of . In this paper we give some results connecting the neighbor-integrity and binary graph operations.