Graph Parameters, Implicit Representations and Factorial Properties

Abstract

How to efficiently represent a graph in computer memory is a fundamental data structuring question. In the present paper, we address this question from a combinatorial point of view. A representation of an n-vertex graph G is called implicit if it assigns to each vertex of G a binary code of length O (log n) so that the adjacency of two vertices is a function of their codes. A necessary condition for a hereditary class X of graphs to admit an implicit representation is that X has at most factorial speed of growth. This condition, however, is not sufficient, as was recently shown in [19]. Several sufficient conditions for the existence of implicit representations deal with boundedness of some parameters, such as degeneracy or clique-width. In the present paper, we analyze more graph parameters and prove a number of new results related to implicit representation and factorial properties.

Graph parameters, implicit representations and factorial properties

“…This extends to a transformation between hereditary classes: transform every graph in a hereditary class to a bipartite graph and take the hereditary closure of the obtained set of bipartite graphs. As was shown in [18], this transformation preserves the factorial speed of growth as well as the existence of an implicit representation 1 .…”

“…Since v is not the centre of a P 5 , the graph G 1 is a chain graph, and hence admits an implicit representation. For i > 1, we show that the graph G i does not contain a onesided copy of F 1 1,k with the vertex of large degree in V i . Indeed, assume that G i contains a one-sided copy of F 1 1,k with the vertex of large degree in V i , and denote the two vertices of this copy in V i by a and b.…”

“…For i > 1, we show that the graph G i does not contain a onesided copy of F 1 1,k with the vertex of large degree in V i . Indeed, assume that G i contains a one-sided copy of F 1 1,k with the vertex of large degree in V i , and denote the two vertices of this copy in V i by a and b. By definition, a and b have neighbours in V i−1 .…”

Optimal Adjacency Labels for Subgraphs of Cartesian Products

“…In this paper, we present optimal adjacency labeling schemes (equivalently, induced-universal graph constructions) for subgraphs of Cartesian products, which essentially closes a recent line of work studying these objects [1,2,3,4,8,10].…”

“…Write mon(F □ ) and her(F □ ), respectively, for the monotone and hereditary closures of this class, which are the sets of all graphs G that are a subgraph (respectively, induced subgraph) of some H ∈ F □ . We will construct optimal labeling schemes for mon(F □ ) and her(F □ ) from an optimal labeling scheme for F. Cartesian products appear several times independently in the recent literature on labeling schemes [3,8,2] (and later in [10,1,4]), and are extremely natural for the problem of adjacency labeling for a few reasons.…”