Displaying similar documents to “Minimal reducible bounds for hom-properties of graphs”

Minimal forbidden subgraphs of reducible graph properties

Amelie J. Berger (2001)

Discussiones Mathematicae Graph Theory

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A property of graphs is any class of graphs closed under isomorphism. Let ₁,₂,...,ₙ be properties of graphs. A graph G is (₁,₂,...,ₙ)-partitionable if the vertex set V(G) can be partitioned into n sets, V₁,V₂,..., Vₙ, such that for each i = 1,2,...,n, the graph G [ V i ] ∈ i . We write ₁∘₂∘...∘ₙ for the property of all graphs which have a (₁,₂,...,ₙ)-partition. An additive induced-hereditary property is called reducible if there exist additive induced-hereditary properties ₁ and ₂ such that = ₁∘₂....

𝓟-bipartitions of minor hereditary properties

Piotr Borowiecki, Jaroslav Ivančo (1997)

Discussiones Mathematicae Graph Theory

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We prove that for any two minor hereditary properties 𝓟₁ and 𝓟₂, such that 𝓟₂ covers 𝓟₁, and for any graph G ∈ 𝓟₂ there is a 𝓟₁-bipartition of G. Some remarks on minimal reducible bounds are also included.

A survey of hereditary properties of graphs

Mieczysław Borowiecki, Izak Broere, Marietjie Frick, Peter Mihók, Gabriel Semanišin (1997)

Discussiones Mathematicae Graph Theory

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In this paper we survey results and open problems on the structure of additive and hereditary properties of graphs. The important role of vertex partition problems, in particular the existence of uniquely partitionable graphs and reducible properties of graphs in this structure is emphasized. Many related topics, including questions on the complexity of related problems, are investigated.

Unique factorization theorem for object-systems

Peter Mihók, Gabriel Semanišin (2011)

Discussiones Mathematicae Graph Theory

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The concept of an object-system is a common generalization of simple graph, digraph and hypergraph. In the theory of generalised colourings of graphs, the Unique Factorization Theorem (UFT) for additive induced-hereditary properties of graphs provides an analogy of the well-known Fundamental Theorem of Arithmetics. The purpose of this paper is to present UFT for object-systems. This result generalises known UFT for additive induced-hereditary and hereditary properties of graphs and digraphs....

Prime ideals in the lattice of additive induced-hereditary graph properties

Amelie J. Berger, Peter Mihók (2003)

Discussiones Mathematicae Graph Theory

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An additive induced-hereditary property of graphs is any class of finite simple graphs which is closed under isomorphisms, disjoint unions and induced subgraphs. The set of all additive induced-hereditary properties of graphs, partially ordered by set inclusion, forms a completely distributive lattice. We introduce the notion of the join-decomposability number of a property and then we prove that the prime ideals of the lattice of all additive induced-hereditary properties are divided...

Further results on sequentially additive graphs

Suresh Manjanath Hegde, Mirka Miller (2007)

Discussiones Mathematicae Graph Theory

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Given a graph G with p vertices, q edges and a positive integer k, a k-sequentially additive labeling of G is an assignment of distinct numbers k,k+1,k+2,...,k+p+q-1 to the p+q elements of G so that every edge uv of G receives the sum of the numbers assigned to the vertices u and v. A graph which admits such an assignment to its elements is called a k-sequentially additive graph. In this paper, we give an upper bound for k with respect to which the given graph may possibly be k-sequentially...

On universal graphs for hom-properties

Peter Mihók, Jozef Miškuf, Gabriel Semanišin (2009)

Discussiones Mathematicae Graph Theory

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A graph property is any isomorphism closed class of simple graphs. For a simple finite graph H, let → H denote the class of all simple countable graphs that admit homomorphisms to H, such classes of graphs are called hom-properties. Given a graph property 𝓟, a graph G ∈ 𝓟 is universal in 𝓟 if each member of 𝓟 is isomorphic to an induced subgraph of G. In particular, we consider universal graphs in → H and we give a new proof of the existence of a universal graph in → H, for any finite...

Requiring that Minimal Separators Induce Complete Multipartite Subgraphs

Terry A. McKee (2018)

Discussiones Mathematicae Graph Theory

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Complete multipartite graphs range from complete graphs (with every partite set a singleton) to edgeless graphs (with a unique partite set). Requiring minimal separators to all induce one or the other of these extremes characterizes, respectively, the classical chordal graphs and the emergent unichord-free graphs. New theorems characterize several subclasses of the graphs whose minimal separators induce complete multipartite subgraphs, in particular the graphs that are 2-clique sums...

On some variations of extremal graph problems

Gabriel Semanišin (1997)

Discussiones Mathematicae Graph Theory

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A set P of graphs is termed hereditary property if and only if it contains all subgraphs of any graph G belonging to P. A graph is said to be maximal with respect to a hereditary property P (shortly P-maximal) whenever it belongs to P and none of its proper supergraphs of the same order has the property P. A graph is P-extremal if it has a the maximum number of edges among all P-maximal graphs of given order. The number of its edges is denoted by ex(n, P). If the number of edges...

Universality for and in Induced-Hereditary Graph Properties

Izak Broere, Johannes Heidema (2013)

Discussiones Mathematicae Graph Theory

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The well-known Rado graph R is universal in the set of all countable graphs I, since every countable graph is an induced subgraph of R. We study universality in I and, using R, show the existence of 2 א0 pairwise non-isomorphic graphs which are universal in I and denumerably many other universal graphs in I with prescribed attributes. Then we contrast universality for and universality in induced-hereditary properties of graphs and show that the overwhelming majority of induced-hereditary...

The Quest for A Characterization of Hom-Properties of Finite Character

Izak Broere, Moroli D.V. Matsoha, Johannes Heidema (2016)

Discussiones Mathematicae Graph Theory

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A graph property is a set of (countable) graphs. A homomorphism from a graph G to a graph H is an edge-preserving map from the vertex set of G into the vertex set of H; if such a map exists, we write G → H. Given any graph H, the hom-property →H is the set of H-colourable graphs, i.e., the set of all graphs G satisfying G → H. A graph property P is of finite character if, whenever we have that F ∈ P for every finite induced subgraph F of a graph G, then we have that G ∈ P too. We explore...

Product rosy labeling of graphs

Dalibor Fronček (2008)

Discussiones Mathematicae Graph Theory

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In this paper we describe a natural extension of the well-known ρ-labeling of graphs (also known as rosy labeling). The labeling, called product rosy labeling, labels vertices with elements of products of additive groups. We illustrate the usefulness of this labeling by presenting a recursive construction of infinite families of trees decomposing complete graphs.

A cancellation property for the direct product of graphs

Richard H. Hammack (2008)

Discussiones Mathematicae Graph Theory

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Given graphs A, B and C for which A×C ≅ B×C, it is not generally true that A ≅ B. However, it is known that A×C ≅ B×C implies A ≅ B provided that C is non-bipartite, or that there are homomorphisms from A and B to C. This note proves an additional cancellation property. We show that if B and C are bipartite, then A×C ≅ B×C implies A ≅ B if and only if no component of B admits an involution that interchanges its partite sets.

Graphs maximal with respect to hom-properties

Jan Kratochvíl, Peter Mihók, Gabriel Semanišin (1997)

Discussiones Mathematicae Graph Theory

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For a simple graph H, →H denotes the class of all graphs that admit homomorphisms to H (such classes of graphs are called hom-properties). We investigate hom-properties from the point of view of the lattice of hereditary properties. In particular, we are interested in characterization of maximal graphs belonging to →H. We also provide a description of graphs maximal with respect to reducible hom-properties and determine the maximum number of edges of graphs belonging to →H.

On the lattice of additive hereditary properties of finite graphs

Ján Jakubík (2002)

Discussiones Mathematicae - General Algebra and Applications

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In this paper it is proved that the lattice of additive hereditary properties of finite graphs is completely distributive and that it does not satisfy the Jordan-Dedekind condition for infinite chains.