On a Vizing-like Conjecture for Direct Product Graphs

Let fl(G) be the domination number of a graph G, and let G \Theta H be the direct product of graphs G and H . It is shown that for any k 0 there exists a graph G such that fl(G \Theta G) fl(G

ON DOMINATION NUMBERS OF GRAPH BUNDLES

Let γ(G) be the domination number of a graph G. It is shown that for any k ≥ 0 there exists a Cartesian graph bundle B✷ϕF such that γ(B✷ϕF)=γ(B)γ(F)−2k. The domination numbers of Cartesian bundles of two cycles are determined exactly when the fibre graph is a triangle or a square. A statement similar to Vizing’s conjecture on strong graph bundles is shown not to be true by proving the inequality γ(B ✷ × ϕF) ≤ γ(B)γ(F) for strong graph bundles. Examples of graphs B and F with γ(B ✷ × ϕF) <γ(B)γ(F) are given

Recognizing weighted directed cartesian graph bundles

In this paper we show that methods for recognizing Cartesian graph bundles can be generalized to weighted digraphs. The main result is an algorithm which lists the sets of degenerate arcs for all representations of digraph as a weighted directed Cartesian graph bundle over simple base digraphs not containing transitive tournament on three vertices. Two main notions are used. The first one is the new relation $^→δ*$defined among the arcs of a digraph as a weighted directed analogue of the well-known relation δ*. The second one is the concept of half-convex subgraphs. A subgraph H is half-convex in G if any vertex x ∈ G∖H has at most one predecessor and at most one successor

Weak k-reconstruction of Cartesian products

By Ulam's conjecture every finite graph G can be reconstructed from its deck of vertex deleted subgraphs. The conjecture is still open, but many special cases have been settled. In particular, one can reconstruct Cartesian products. We consider the case of k-vertex deleted subgraphs of Cartesian products, and prove that one can decide whether a graph H is a k-vertex deleted subgraph of a Cartesian product G with at least k+1 prime factors on at least k+1 vertices each, and that H uniquely determines G. This extends previous work of the authors and Sims. The paper also contains a counterexample to a conjecture of MacAvaney