On prime inductive classes of graphs

Discussiones Mathematicae Graph Theory, 2002

A natural generalization of the fundamental graph vertex-colouring problem leads to the class of problems known as generalized or improper colourings. These problems can be very well described in the language of reducible (induced) hereditary properties of graphs. It turned out that a very useful tool for the unique determination of these properties are generating sets. In this paper we focus on the structure of specific generating sets which provide the base for the proof of The Unique Factorization Theorem for induced-hereditary properties of graphs.

In graph theory, the Helly property has been applied to families of sets, such as cliques, disks, bicliques, and neighbourhoods, leading to the classes of clique-Helly, disk-Helly, biclique-Helly, neighbourhood-Helly graphs, respectively. A natural question is to determine for which graphs the corresponding Helly property holds, for every induced subgraph. This leads to the corresponding classes of hereditary clique-Helly, hereditary disk-Helly, hereditary biclique-Helly and hereditary neighbourhood-Helly graphs. In this paper, we describe characterizations in terms of families of forbidden subgraphs, for the classes of hereditary biclique-Helly and hereditary neighbourhood-Helly graphs. We consider both open and closed neighbourhoods. The forbidden subgraphs are all of fixed size, implying polynomial time recognition for these classes.

LATIN 2020: Theoretical Informatics, 2020

We consider the classical problems (Edge) Steiner Tree and Vertex Steiner Tree after restricting the input to some class of graphs characterized by a small set of forbidden induced subgraphs. We show a dichotomy for the former problem restricted to (H1, H2)free graphs and a dichotomy for the latter problem restricted to H-free graphs. We find that there exists an infinite family of graphs H such that Vertex Steiner Tree is polynomial-time solvable for H-free graphs, whereas there exist only two graphs H for which this holds for Edge Steiner Tree. We also find that Edge Steiner Tree is polynomialtime solvable for (H1, H2)-free graphs if and only if the treewidth of the class of (H1, H2)-free graphs is bounded (subject to P = NP). To obtain the latter result, we determine all pairs (H1, H2) for which the class of (H1, H2)-free graphs has bounded treewidth. Supported by the Leverhulme Trust (RPG-2016-258) and the Royal Society (IES\R1\191223). Open Problem 3 Does there exist a pair (H 1 , H 2) such that Vertex Steiner Tree and unweighted Vertex Steiner Tree have different complexities for (H 1 , H 2)-free graphs? Open Problem 4 For every integer t, determine the complexity of Vertex Steiner Tree for (K 1,3 , P t)-free graphs. To obtain an answer to Open Problem 4, we need new insights into the structure of (K 1,3 , P t)-free graphs. These insights may also be useful to obtain new results for other problems, such as the Graph Colouring problem restricted to (K 1,3 , P t)-free graphs (see [11,13]).

Journal of Computer and System Sciences, 2008

This paper investigates a graph enumeration problem, called the maximal P-subgraphs problem, where P is a hereditary or connected-hereditary graph property. Formally, given a graph G, the maximal P-subgraphs problem is to generate all maximal induced subgraphs of G that satisfy P. This problem differs from the well-known node-deletion problem, studied by Yannakakis and Lewis [J. Lewis, On the complexity of the maximum subgraph J. Lewis, M. Yannakakis, The node-deletion problem for hereditary properties is NP-complete, J. Comput. System Sci. 20 (2) (1980) 219-230]. In the maximal P-subgraphs problem, the goal is to produce all (locally) maximal subgraphs of a graph that have property P, whereas in the node-deletion problem, the goal is to find a single (globally) maximum size subgraph with property P. Algorithms are presented that reduce the maximal P-subgraphs problem to an input-restricted version of this problem. These algorithms imply that when attempting to efficiently solve the maximal P-subgraphs problem for a specific P, it is sufficient to solve the restricted case. The main contributions of this paper are characterizations of when the maximal P-subgraphs problem is in a complexity class C (e.g., polynomial delay, total polynomial time).

European Journal of Combinatorics, 2014

The class of split graphs consists of those graphs G, for which there exist partitions (V 1 , V 2) of their vertex sets V (G) such that G[V 1 ] is an edgeless graph and G[V 2 ] is a complete graph. The classes of edgeless and complete graphs are members of a family L * ≤ , which consists of all graph classes that are induced hereditary and closed under substitution. Graph classes considered in this paper are alike to split graphs. Namely a graph class P is an object of our interest if there exist two graph classes P 1 , P 2 ∈ L * ≤ (not necessarily different) such that for each G ∈ P we can find a partition (V 1 , V 2) of V (G) satisfying G[V 1 ] ∈ P 1 and G[V 2 ] ∈ P 2. For each such class P we characterize all forbidden graphs that are strongly decomposable. The finiteness of families of forbidden graphs for P is analyzed giving a result characterizing classes with a finite number of forbidden graphs. Our investigation confirms, in the class L * ≤ , Zverovich's conjecture describing all induced hereditary graph classes defined by generalized vertex 2-partitions that have finite families of forbidden graphs.

2007

An ordered graph is a graph together with a linear order on its vertices. A hereditary property of ordered graphs is a collection of ordered graphs closed under taking order-preserving isomorphisms of the vertex set, and order-preserving induced subgraphs. If P is a hereditary property of ordered graphs, then P n denotes the collection \( \left\{ {G \in \mathcal{P}:V(G) = [n]} \right\} \) , and the function \( n \mapsto \left| {\mathcal{P}_n } \right| \) is called the speed of P. The possible speeds of a hereditary property of labelled graphs have been extensively studied (see [BBW00] and [Bol98] for example), and more recently hereditary properties of other combinatorial structures, such as oriented graphs ([AS00], [BBM06+c]), posets ([BBM06+a], [BGP99]), words ([BB05], [QZ04]) and permutations ([KK03], [MT04]), have also attracted attention. Properties of ordered graphs generalize properties of both labelled graphs and permutations. In this paper we determine the possible speeds of a hereditary property of ordered graphs, up to the speed 2n−1. In particular, we prove that there exists a jump from polynomial speed to speed F n, the Fibonacci numbers, and that there exists an infinite sequence of subsequent jumps, from p(n)F n,k to F n,k+1 (where p(n) is a polynomial and F n,k are the generalized Fibonacci numbers) converging to 2n−1. Our results generalize a theorem of Kaiser and Klazar [KK03], who proved that the same jumps occur for hereditary properties of permutations.

Discussiones Mathematicae Graph Theory, 2018

Let P be an arbitrary class of graphs that is closed under taking induced subgraphs and let C(P) be the family of forbidden subgraphs for P. We investigate the class P(k) consisting of all the graphs G for which the removal of no more than k vertices results in graphs that belong to P. This approach provides an analogy to apex graphs and apex-outerplanar graphs studied previously. We give a sharp upper bound on the number of vertices of graphs in C(P(1)) and we give a construction of graphs in C(P(k)) of relatively large order for k ≥ 2. This construction implies a lower bound on the maximum order of graphs in C(P(k)). Especially, we investigate C(W r (1)), where W r denotes the class of P r-free graphs. We determine some forbidden subgraphs for the class W r (1) with the minimum and maximum number of vertices. Moreover, we give sufficient conditions for graphs belonging to C(P(k)), where P is an additive class, and a characterisation of all forests in C(P(k)). Particularly we deal with C(P(1)), where P is a class closed under substitution and obtain a characterisation of all graphs in the corresponding C(P(1)). In order to obtain desired results we exploit some hypergraph tools and this technique gives a new result in the hypergraph theory.

Discrete Mathematics, 2021

The Ramsey number R X (p, q) for a class of graphs X is the minimum n such that every graph in X with at least n vertices has either a clique of size p or an independent set of size q. We say that Ramsey numbers are linear in X if there is a constant k such that R X (p, q) ≤ k(p + q) for all p, q. In the present paper we conjecture that if X is a hereditary class defined by finitely many forbidden induced subgraphs, then Ramsey numbers are linear in X if and only if X excludes a forest, a disjoint union of cliques and their complements. We prove the "only if" part of this conjecture and verify the "if" part for a variety of classes. We also apply the notion of linearity to bipartite Ramsey numbers and reveal a number of similarities and differences between the bipartite and non-bipartite case.

Journal of Graph Theory, 2005

A graph property (i.e., a set of graphs) is induced-hereditary or ad- ditive if it is closed under taking induced-subgraphs or disjoint unions. If P and Q are properties, the product P ◦ Q consists of all graphs G for which there is a partition of the vertex set of G into (possibly empty) subsets A and B with G(A)

Discussiones Mathematicae Graph Theory, 2009

An additive hereditary graph property is any class of simple graphs, which is closed under isomorphisms unions and taking subgraphs. Let L a denote a class of all such properties. In the paper, we consider Hreducible over L a properties with H being a fixed graph. The finiteness of the sets of all minimal forbidden graphs is analyzed for such properties.