Dominating direct products of graphs

AbstractAn upper bound for the domination number of the direct product of graphs is proved. It in particular implies that for any graphs G and H, γ(G×H)⩽3γ(G)γ(H). Graphs with arbitrarily large domination numbers are constructed for which this bound is attained. Concerning the upper domination number we prove that Γ(G×H)⩾Γ(G)Γ(H), thus confirming a conjecture from [R. Nowakowski, D.F. Rall, Associative graph products and their independence, domination and coloring numbers, Discuss. Math. Graph Theory 16 (1996) 53–79]. Finally, for paired-domination of direct products we prove that γpr(G×H)⩽γpr(G)γpr(H) for arbitrary graphs G and H, and also present some infinite families of graphs that attain this bound

Reasoning about Independence in Probabilistic Models of Relational Data

We extend the theory of d-separation to cases in which data instances are not independent and identically distributed. We show that applying the rules of d-separation directly to the structure of probabilistic models of relational data inaccurately infers conditional independence. We introduce relational d-separation, a theory for deriving conditional independence facts from relational models. We provide a new representation, the abstract ground graph, that enables a sound, complete, and computationally efficient method for answering d-separation queries about relational models, and we present empirical results that demonstrate effectiveness.Comment: 61 pages, substantial revisions to formalisms, theory, and related wor

Vertex decomposable graphs and obstructions to shellability

Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5-chordal graphs with no chordless 4-cycles are shellable and sequentially Cohen-Macaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability.Comment: 13 pages, 3 figures. v2: Improved exposition, added Section 5.2 and additional references. v3: minor corrections for publicatio