Graph parameters, implicit representations and factorial properties

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 0(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 and the speed of hereditary properties

In this thesis we study the speed of hereditary properties of graphs and how this defines some of the structure of the properties. We start by characterizing several graph parameters by means of minimal hereditary classes. We then give a global characterization of properties of low speed, before looking at properties with higher speeds starting at the Bell number. We then introduce a new parameter, clique-width, and show that there are an infinite amount of minimal hereditary properties with unbounded clique-width. We then look at the factorial layer in more detail and focus on P7-free bipartite graphs. Finally we discuss word-representable graphs

Structural solutions to maximum independent set and related problems

In this thesis, we study some fundamental problems in algorithmic graph theory. Most natural problems in this area are hard from a computational point of view. However, many applications demand that we do solve such problems, even if they are intractable. There are a number of methods in which we can try to do this: 1) We may use an approximation algorithm if we do not necessarily require the best possible solution to a problem. 2) Heuristics can be applied and work well enough to be useful for many applications. 3) We can construct randomised algorithms for which the probability of failure is very small. 4) We may parameterize the problem in some way which limits its complexity. In other cases, we may also have some information about the structure of the instances of the problem we are trying to solve. If we are lucky, we may and that we can exploit this extra structure to find efficient ways to solve our problem. The question which arises is - How far must we restrict the structure of our graph to be able to solve our problem efficiently? In this thesis we study a number of problems, such as Maximum Indepen- dent Set, Maximum Induced Matching, Stable-II, Efficient Edge Domina- tion, Vertex Colouring and Dynamic Edge-Choosability. We try to solve problems on various hereditary classes of graphs and analyse the complexity of the resulting problem, both from a classical and parameterized point of view

On factorial properties of chordal bipartite graphs

For a graph property X, let Xn be the number of graphs with vertex set {1,…,n} having property X, also known as the speed of X. A property X is called factorial if X is hereditary (i.e., closed under taking induced subgraphs) and nc1n≤Xn≤nc2n for some positive constants c1 and c2. Hereditary properties with speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored. To better understand the structure of factorial properties we look for minimal superfactorial ones. In [J.P. Spinrad, Nonredundant 1’s in Γ-free matrices, SIAM J. Discrete Math. 8 (1995) 251–257], Spinrad showed that the number of n-vertex chordal bipartite graphs is 2Θ(nlog2n), which means that this class is superfactorial. On the other hand, all subclasses of chordal bipartite graphs that have been studied in the literature, such as forest, bipartite permutation, bipartite distance-hereditary or convex graphs, are factorial. In this paper, we study more hereditary subclasses of chordal bipartite graphs and reveal both factorial and superfactorial members in this family. The latter fact shows that the class of chordal bipartite graphs is not a minimal superfactorial one. Finding minimal superfactorial classes in this family remains a challenging open question