Minimal forbidden subgraphs of reducible graph properties

Amelie J. Berger

Displaying similar documents to “Minimal forbidden subgraphs of reducible graph properties”

Unique factorization theorem

Peter Mihók (2000)

Discussiones Mathematicae Graph Theory

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A property of graphs is any class of graphs closed under isomorphism. A property of graphs is induced-hereditary and additive if it is closed under taking induced subgraphs and disjoint unions of graphs, respectively. Let ₁,₂, ...,ₙ be properties of graphs. A graph G is (₁,₂,...,ₙ)-partitionable (G has property ₁ º₂ º... ºₙ) if the vertex set V(G) of G can be partitioned into n sets V₁,V₂,..., Vₙ such that the subgraph $G [V_{i}]$ of G induced by Vi belongs to $_{i}$ ; i = 1,2,...,n. A property is said...

On a characterization of graphs by average labellings

Matúš Harminc (1997)

Discussiones Mathematicae Graph Theory

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The additive hereditary property of linear forests is characterized by the existence of average labellings.

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

Ewa Drgas-Burchardt (2009)

Discussiones Mathematicae Graph Theory

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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.

Unique factorisation of additive induced-hereditary properties

Alastair Farrugia, R. Bruce Richter (2004)

Discussiones Mathematicae Graph Theory

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An additive hereditary graph property is a set of graphs, closed under isomorphism and under taking subgraphs and disjoint unions. Let ₁,...,ₙ be additive hereditary graph properties. A graph G has property (₁∘...∘ₙ) if there is a partition (V₁,...,Vₙ) of V(G) into n sets such that, for all i, the induced subgraph $G [V_{i}]$ is in $_{i}$ . A property is reducible if there are properties , such that = ∘ ; otherwise it is irreducible. Mihók, Semanišin and Vasky [8] gave a factorisation for any additive...

Generalized edge-chromatic numbers and additive hereditary properties of graphs

Michael J. Dorfling, Samantha Dorfling (2002)

Discussiones Mathematicae Graph Theory

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An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. Let and be hereditary properties of graphs. The generalized edge-chromatic number $ρ_{}^{'} ()$ is defined as the least integer n such that ⊆ n. We investigate the generalized edge-chromatic numbers of the properties → H, ₖ, ₖ, *ₖ, ₖ and ₖ.

Uniquely partitionable graphs

Jozef Bucko, Marietjie Frick, Peter Mihók, Roman Vasky (1997)

Discussiones Mathematicae Graph Theory

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Let ₁,...,ₙ be properties of graphs. A (₁,...,ₙ)-partition of a graph G is a partition of the vertex set V(G) into subsets V₁, ...,Vₙ such that the subgraph $G [V_{i}]$ induced by $V_{i}$ has property $_{i}$ ; i = 1,...,n. A graph G is said to be uniquely (₁, ...,ₙ)-partitionable if G has exactly one (₁,...,ₙ)-partition. A property is called hereditary if every subgraph of every graph with property also has property . If every graph that is a disjoint union of two graphs that have property also has property...

On Ramsey $(K_{1, 2}, K ₙ)$ -minimal graphs

Mariusz Hałuszczak (2012)

Discussiones Mathematicae Graph Theory

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Let F be a graph and let , denote nonempty families of graphs. We write F → (,) if in any 2-coloring of edges of F with red and blue, there is a red subgraph isomorphic to some graph from G or a blue subgraph isomorphic to some graph from H. The graph F without isolated vertices is said to be a (,)-minimal graph if F → (,) and F - e not → (,) for every e ∈ E(F). We present a technique which allows to generate infinite family of (,)-minimal graphs if we know some special graphs. In particular,...

Minimal reducible bounds for hom-properties of graphs

Amelie Berger, Izak Broere (1999)

Discussiones Mathematicae Graph Theory

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Let H be a fixed finite graph and let → H be a hom-property, i.e. the set of all graphs admitting a homomorphism into H. We extend the definition of → H to include certain infinite graphs H and then describe the minimal reducible bounds for → H in the lattice of additive hereditary properties and in the lattice of hereditary properties.

Maximal graphs with respect to hereditary properties

Izak Broere, Marietjie Frick, Gabriel Semanišin (1997)

Discussiones Mathematicae Graph Theory

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A property of graphs is a non-empty set of graphs. A property P is called hereditary if every subgraph of any graph with property P also has property P. Let P₁, ...,Pₙ be properties of graphs. We say that a graph G has property P₁∘...∘Pₙ if the vertex set of G can be partitioned into n sets V₁, ...,Vₙ such that the subgraph of G induced by Vi has property $P_{i}$ ; i = 1,..., n. A hereditary property R is said to be reducible if there exist two hereditary properties P₁ and P₂ such that R =...

New results on generalized graph coloring.

Alekseev, Vladimir E., Farrugia, Alastair, Lozin, Vadim V. (2004)

Discrete Mathematics and Theoretical Computer Science. DMTCS [electronic only]

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On hereditary properties of composition graphs

Vadim E. Levit, Eugen Mandrescu (1998)

Discussiones Mathematicae Graph Theory

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The composition graph of a family of n+1 disjoint graphs $H_{i} : 0 \leq i \leq n$ is the graph H obtained by substituting the n vertices of H₀ respectively by the graphs H₁,H₂,...,Hₙ. If H has some hereditary property P, then necessarily all its factors enjoy the same property. For some sort of graphs it is sufficient that all factors $H_{i} : 0 \leq i \leq n$ have a certain common P to endow H with this P. For instance, it is known that the composition graph of a family of perfect graphs is also a perfect graph (B. Bollobas, 1978),...

Uniquely partitionable planar graphs with respect to properties having a forbidden tree

Jozef Bucko, Jaroslav Ivančo (1999)

Discussiones Mathematicae Graph Theory

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Let ₁, ₂ be graph properties. A vertex (₁,₂)-partition of a graph G is a partition V₁,V₂ of V(G) such that for i = 1,2 the induced subgraph $G [V_{i}]$ has the property $_{i}$ . A property ℜ = ₁∘₂ is defined to be the set of all graphs having a vertex (₁,₂)-partition. A graph G ∈ ₁∘₂ is said to be uniquely (₁,₂)-partitionable if G has exactly one vertex (₁,₂)-partition. In this note, we show the existence of uniquely partitionable planar graphs with respect to hereditary additive properties having a...

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.

On infinite uniquely partitionable graphs and graph properties of finite character

Jozef Bucko, Peter Mihók (2009)

Discussiones Mathematicae Graph Theory

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A graph property is any nonempty isomorphism-closed class of simple (finite or infinite) graphs. A graph property is of finite character if a graph G has a property if and only if every finite induced subgraph of G has a property . Let ₁,₂,...,ₙ be graph properties of finite character, a graph G is said to be (uniquely) (₁, ₂, ...,ₙ)-partitionable if there is an (exactly one) partition V₁, V₂, ..., Vₙ of V(G) such that $G [V_{i}] \in_{i}$ for i = 1,2,...,n. Let us denote by ℜ = ₁ ∘ ₂ ∘ ... ∘ ₙ the class...

Reducible properties of graphs

P. Mihók, G. Semanišin (1995)

Discussiones Mathematicae Graph Theory

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Let L be the set of all hereditary and additive properties of graphs. For P₁, P₂ ∈ L, the reducible property R = P₁∘P₂ is defined as follows: G ∈ R if and only if there is a partition V(G) = V₁∪ V₂ of the vertex set of G such that $⟨ V ₁ ⟩_{G} \in P ₁$ and $⟨ V ₂ ⟩_{G} \in P ₂$ . The aim of this paper is to investigate the structure of the reducible properties of graphs with emphasis on the uniqueness of the decomposition of a reducible property into irreducible ones.

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....

Criteria for of the existence of uniquely partitionable graphs with respect to additive induced-hereditary properties

Izak Broere, Jozef Bucko, Peter Mihók (2002)

Discussiones Mathematicae Graph Theory

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Let ₁,₂,...,ₙ be graph properties, a graph G is said to be uniquely (₁,₂, ...,ₙ)-partitionable if there is exactly one (unordered) partition V₁,V₂,...,Vₙ of V(G) such that $G [V_{i}] \in_{i}$ for i = 1,2,...,n. We prove that for additive and induced-hereditary properties uniquely (₁,₂,...,ₙ)-partitionable graphs exist if and only if $_{i}$ and $_{j}$ are either coprime or equal irreducible properties of graphs for every i ≠ j, i,j ∈ 1,2,...,n.

Minimal rankings of the Cartesian product Kₙ ☐ Kₘ

Gilbert Eyabi, Jobby Jacob, Renu C. Laskar, Darren A. Narayan, Dan Pillone (2012)

Discussiones Mathematicae Graph Theory

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For a graph G = (V, E), a function f:V(G) → 1,2, ...,k is a k-ranking if f(u) = f(v) implies that every u - v path contains a vertex w such that f(w) > f(u). A k-ranking is minimal if decreasing any label violates the definition of ranking. The arank number, $ψ_{r} (G)$ , of G is the maximum value of k such that G has a minimal k-ranking. We completely determine the arank number of the Cartesian product Kₙ ☐ Kₙ, and we investigate the arank number of Kₙ ☐ Kₘ where n > m.

Clopen graphs

Stefan Geschke (2013)

Fundamenta Mathematicae

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A graph G on a topological space X as its set of vertices is clopen if the edge relation of G is a clopen subset of X² without the diagonal. We study clopen graphs on Polish spaces in terms of their finite induced subgraphs and obtain information about their cochromatic numbers. In this context we investigate modular profinite graphs, a class of graphs obtained from finite graphs by taking inverse limits. This continues the investigation of continuous colorings on Polish spaces and their...

Generalized chromatic numbers and additive hereditary properties of graphs

Izak Broere, Samantha Dorfling, Elizabeth Jonck (2002)

Discussiones Mathematicae Graph Theory

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An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. Let and be additive hereditary properties of graphs. The generalized chromatic number $χ_{} ()$ is defined as follows: $χ_{} () = n$ iff ⊆ ⁿ but $⊊^{n - 1}$ . We investigate the generalized chromatic numbers of the well-known properties of graphs ₖ, ₖ, ₖ, ₖ and ₖ.

Structure of the set of all minimal total dominating functions of some classes of graphs

K. Reji Kumar, Gary MacGillivray (2010)

Discussiones Mathematicae Graph Theory

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In this paper we study some of the structural properties of the set of all minimal total dominating functions ( $_{T}$ ) of cycles and paths and introduce the idea of function reducible graphs and function separable graphs. It is proved that a function reducible graph is a function separable graph. We shall also see how the idea of function reducibility is used to study the structure of $_{T} (G)$ for some classes of graphs.

On 2-periodic graphs of a certain graph operator

Ivan Havel, Bohdan Zelinka (2001)

Discussiones Mathematicae Graph Theory

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We deal with the graph operator $\bar{P o w ₂}$ defined to be the complement of the square of a graph: $\bar{P o w ₂} (G) = \bar{P o w ₂ (G)}$ . Motivated by one of many open problems formulated in [6] we look for graphs that are 2-periodic with respect to this operator. We describe a class of bipartite graphs possessing the above mentioned property and prove that for any m,n ≥ 6, the complete bipartite graph $K_{m, n}$ can be decomposed in two edge-disjoint factors from . We further show that all the incidence graphs of Desarguesian finite projective...

Digraphs with isomorphic underlying and domination graphs: connected $U G^{c} (d)$

Kim A.S. Factor, Larry J. Langley (2007)

Discussiones Mathematicae Graph Theory

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The domination graph of a directed graph has an edge between vertices x and y provided either (x,z) or (y,z) is an arc for every vertex z distinct from x and y. We consider directed graphs D for which the domination graph of D is isomorphic to the underlying graph of D. We demonstrate that the complement of the underlying graph must have k connected components isomorphic to complete graphs, paths, or cycles. A complete characterization of directed graphs where k = 1 is presented. ...

Graphs with small additive stretch number

Dieter Rautenbach (2004)

Discussiones Mathematicae Graph Theory

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The additive stretch number $s_{a d d} (G)$ of a graph G is the maximum difference of the lengths of a longest induced path and a shortest induced path between two vertices of G that lie in the same component of G.We prove some properties of minimal forbidden configurations for the induced-hereditary classes of graphs G with $s_{a d d} (G) \leq k$ for some k ∈ N₀ = 0,1,2,.... Furthermore, we derive characterizations of these classes for k = 1 and k = 2.

Symmetries of embedded complete bipartite graphs

Erica Flapan, Nicole Lehle, Blake Mellor, Matt Pittluck, Xan Vongsathorn (2014)

Fundamenta Mathematicae

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We characterize which automorphisms of an arbitrary complete bipartite graph $K_{n, m}$ can be induced by a homeomorphism of some embedding of the graph in S³.