The disjoint domination number γγ(G) of a graph G is the minimum cardinality of the union of two disjoint dominating sets in G. The disjoint independent domination number of a graph G is the minimum cardinality of the union of two disjoint independent dominating sets in G. In this paper we study these two parameters. We determine the value of γγ(G) for several graphs and give partial answers to...

In the previous lectures, we have defined the Shannon Capacity of a graph, Θ(G). This lecture is concerned with the Shannon Capacity of the disjoint union G +H of two graphs G and H. The union of two graphs is, informally, what you get when you just place one graph next to the other. In 1956, Shannon conjectured [1] that Θ(G + H) = Θ(G) + Θ(H). As we will see, this is a very natural conjecture ...

We propose a polynomial-time algorithm which takes as input a finite set of points of R3 and compute, up to arbitrary precision, a maximum subset with diameter at most 1. More precisely, we give the first randomized EPTAS and deterministic PTAS for Maximum Clique in unit ball graphs. Our approximation algorithm also works on disk graphs with arbitrary radii. Almost three decades ago, an elegant...

Abstract. Many “good” topologies for interconnection networks are based on line digraphs of regular digraphs. These digraphs support unitary matrices. We propose the property “being the digraph of a unitary matrix” as additional criterion for the design of new interconnection networks. We define a composition of digraphs, which we call diagonal union. Diagonal union can be used to construct dig...

Let G be a graph of size n with vertex set V (G) and edge set E(G). A ρlabeling of G is a one-to-one function h : V (G) → {0, 1, . . . , 2n} such that {min{|h(u)−h(v)|, 2n+1−|h(u)−h(v)|} : {u, v} ∈ E(G)} = {1, 2, . . . , n}. Such a labeling of G yields a cyclicG-decomposition ofK2n+1. It is known that 2-regular bipartite graphs, the vertex-disjoint union of C3’s, and the vertex-disjoint union o...