Largest reduced neighborhood clique cover number revisited

Let $G$ be a graph and $t\ge 0$. The largest reduced neighborhood clique cover number of $G$, denoted by ${\hat\beta}_t(G)$, is the largest, overall $t$-shallow minors $H$ of $G$, of the smallest number of cliques that can cover any closed neighborhood of a vertex in $H$. It is known that ${\hat\beta}_t(G)\le s_t$, where $G$ is an incomparability graph and $s_t$ is the number of leaves in a largest $t-$shallow minor which is isomorphic to an induced star on $s_t$ leaves. In this paper we give an overview of the properties of ${\hat\beta}_t(G)$ including the connections to the greatest reduced average density of $G$, or $\bigtriangledown_t(G)$, introduce the class of graphs with bounded neighborhood clique cover number, and derive a simple lower and an upper bound for this important graph parameter. We announce two conjectures, one for the value of ${\hat\beta}_t(G)$, and another for a separator theorem (with respect to a certain measure) for an interesting class of graphs, namely the class of incomparability graphs which we suspect to have a polynomial bounded neighborhood clique cover number, when the size of a largest induced star is bounded.Comment: The results in this paper were presented in 48th Southeastern Conference in Combinatorics, Graph Theory and Computing, Florida Atlantic University, Boca Raton, March 201

Representation of Short Distances in Structurally Sparse Graphs

A partial orientation H of a graph G is a weak r-guidance system if for any two vertices at distance at most r in G, there exists a shortest path P between them such that H directs all but one edge in P towards this edge. In case that H has bounded maximum outdegree ?, this gives an efficient representation of shortest paths of length at most r in G: For any pair of vertices, we can either determine the distance between them or decide the distance is more than r, and in the former case, find a shortest path between them, in time O(?^r). We show that graphs from many natural graph classes admit such weak guidance systems, and study the algorithmic aspects of this notion. We also apply the notion to obtain approximation algorithms for distance variants of the independence and domination number in graph classes that admit weak guidance systems of bounded maximum outdegree

Graph Theory

This workshop focused on recent developments in graph theory. These included in particular recent breakthroughs on nowhere-zero flows in graphs, width parameters, applications of graph sparsity in algorithms, and matroid structure results