On prime inductive classes of graphs

AbstractLet H[G1,…,Gn] denote a graph formed from unlabelled graphs G1,…,Gn and a labelled graph H=({v1,…,vn},E) replacing every vertex vi of H by the graph Gi and joining the vertices of Gi with all the vertices of those of Gj whenever {vi,vj}∈E(H). For unlabelled graphs G1,…,Gn,H, let φH(G1,…,Gn) stand for the class of all graphs H[G1,…,Gn] taken over all possible orderings of V(H).A prime inductive class of graphs, I(B,C), is said to be a set of all graphs, which can be produced by recursive applying of φH(G1,…,G∣V(H)∣) where H is a graph from a fixed set C of prime graphs and G1,…,G∣V(H)∣ are either graphs from the set B of prime graphs or graphs obtained in the previous steps. Similar inductive definitions for cographs, k-trees, series–parallel graphs, Halin graphs, bipartite cubic graphs or forbidden structures of some graph classes were considered in the literature (Batagelj (1994) [1] Drgas-Burchardt et al. (2010) [6] and Hajós (1961) [10]).This paper initiates a study of prime inductive classes of graphs giving a result, which characterizes, in their language, the substitution closed induced hereditary graph classes. Moreover, for an arbitrary induced hereditary graph class P it presents a method for the construction of maximal induced hereditary graph classes contained in P and substitution closed.The main contribution of this paper is to give a minimal forbidden graph characterization of induced hereditary prime inductive classes of graphs. As a consequence, the minimal forbidden graph characterization for some special induced hereditary prime inductive graph classes is givenThere is also offered an algebraic view on the class of all prime inductive classes of graphs of the type I({K1},C)

Acyclic homomorphisms to stars of graph Cartesian products and chordal bipartite graphs

AbstractHomomorphisms to a given graph H (H-colourings) are considered in the literature among other graph colouring concepts. We restrict our attention to a special class of H-colourings, namely H is assumed to be a star. Our additional requirement is that the set of vertices of a graph G mapped into the central vertex of the star and any other colour class induce in G an acyclic subgraph. We investigate the existence of such a homomorphism to a star of given order. The complexity of this problem is studied. Moreover, the smallest order of a star for which a homomorphism of a given graph G with desired features exists is considered. Some exact values and many bounds of this number for chordal bipartite graphs, cylinders, grids, in particular hypercubes, are given. As an application of these results, we obtain some bounds on the cardinality of the minimum feedback vertex set for specified graph classes

Minimal vertex Ramsey graphs and minimal forbidden subgraphs

AbstractLet P be a property of graphs. A graph G is vertex (P,k)-colourable if the vertex set V(G) of G can be partitioned into k sets V1,V2,…,Vk such that the subgraph G[Vi] of G belongs to P, i=1,2,…,k. If P is a hereditary property, then the set of minimal forbidden subgraphs of P is defined as follows: F(P)={G:G∉PbuteachpropersubgraphHofGbelongstoP}. In this paper we investigate the property On: each component of G has at most n+1 vertices. We construct minimal forbidden subgraphs for the property (Onk) “to be (On,k)-colourable”.We write G→v(H)k, k⩾2, if for each k-colouring V1,V2,…,Vk of a graph G there exists i, 1⩽i⩽k, such that the graph induced by the set Vi contains H as a subgraph. A graph G is called (H)k-vertex Ramsey minimal if G→v(H)k, but G′↛v(H)k for any proper subgraph G′ of G. The class of (P3)k-vertex Ramsey minimal graphs is investigated

Corrigendum to "Acyclic sum-list-colouring of grids and other classes of graphs" [Opuscula Math. 37, no. 4 (2017), 535-556]

This note provides some minor corrections to the article [Acyclic sum-list-colouring of grids and other classes of graphs, Opuscula Math. 37, no. 4 (2017), 535-556]

A note on joins of additive hereditary graph properties

Let $L^a$ denote a set of additive hereditary graph properties. It is a known fact that a partially ordered set $(L^a, ⊆ )$ is a complete distributive lattice. We present results when a join of two additive hereditary graph properties in $(L^a, ⊆ )$ has a finite or infinite family of minimal forbidden subgraphs

The Number of Stable Matchings in Models of the Gale-Shapley Type with Preferences Given by Partial Orders

From the famous Gale-Shapley theorem we know that each classical marriage problem admits at least one stable matching. This fact has inspired researchers to search for the maximum number of possible stable matchings, which is equivalent to finding the minimum number of unstable matchings among all such problems of size n. In this paper, we deal with this issue for the Gale-Shapley model with preferences represented by arbitrary partial orders. Also, we discuss this model in the context of the classical Gale-Shapley model. (original abstract

Cardinality of a minimal forbidden graph family for reducible additive hereditary graph properties

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 H-reducible 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