Three ways to cover a graph

Abstract

International audienceWe consider the problem of covering an input graph H with graphs from a fixed covering class G. The classical covering number of H with respect to G is the minimum number of graphs from G needed to cover the edges of H without covering non-edges of H. We introduce a unifying notion of three covering parameters with respect to G, two of which are novel concepts only considered in special cases before: the local and the folded covering number. Each parameter measures " how far " H is from G in a different way. Whereas the folded covering number has been investigated thoroughly for some covering classes, e.g., interval graphs and planar graphs, the local covering number has received little attention. We provide new bounds on each covering number with respect to the following covering classes: linear forests, star forests, caterpillar forests, and interval graphs. The classical graph parameters that result this way are interval number, track number, linear arboricity, star arboricity, and caterpillar arboricity. As input graphs we consider graphs of bounded degeneracy, bounded degree, bounded tree-width or bounded simple tree-width, as well as outerplanar, planar bipartite, and planar graphs. For several pairs of an input class and a covering class we determine exactly the maximum ordinary, local, and folded covering number of an input graph with respect to that covering class

On difference graphs and the local dimension of posets

“…We investigate a variant, called the local dimension, which was defined by Ueckerdt [21] and shared with the participants of the Order and Geometry Workshop held in Gu ltowy, Poland, September [14][15][16]2016. The definition was inspired by concepts studied in [3,18]. Definition 1.…”

Local and Union Page Numbers

“…Motivation. Local and union page numbers are motivated by local and union covering numbers as introduced by Knauer and the second author [19]. In order to give a brief summary of the covering number framework, consider a graph H and a graph class G. An injective G-cover of H is a set S = {G 1 , .…”

Local and union boxicity

“…Most recently, Knauer and Ueckerdt [11] suggested the following unifying framework for three kinds of covering numbers, differing in the underlying notion of covering. A graph homomorphism is a map ϕ ∶ V (G) → V (H) with the property that if uv ∈ E(G) then ϕ(u)ϕ(v) ∈ E(H), i.e., ϕ maps vertices of G (not necessarily injectively) to vertices of H such that edges are mapped to edges.…”

“…Proposition 2 (Knauer-Ueckerdt [11]). For every input graph H and every covering class G we have Basically, Theorem 3 says that if H c is not a co-interval graph, there is no way to obtain a co-interval graph from H c by vertex splits.…”

“…Boxicity variants. Let us put the boxicity into the graph covering framework by Knauer and Ueckerdt[11] as described above. To this end, let C denote the…”