Displaying similar documents to “𝓟-bipartitions of minor hereditary properties”

On generating sets of induced-hereditary properties

Gabriel Semanišin (2002)

Discussiones Mathematicae Graph Theory

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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...

Generalized domination, independence and irredudance in graphs

Mieczysław Borowiecki, Danuta Michalak, Elżbieta Sidorowicz (1997)

Discussiones Mathematicae Graph Theory

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The purpose of this paper is to present some basic properties of 𝓟-dominating, 𝓟-independent, and 𝓟-irredundant sets in graphs which generalize well-known properties of dominating, independent and irredundant sets, respectively.

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...

Dominant-matching graphs

Igor' E. Zverovich, Olga I. Zverovich (2004)

Discussiones Mathematicae Graph Theory

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We introduce a new hereditary class of graphs, the dominant-matching graphs, and we characterize it in terms of forbidden induced subgraphs.

On partitions of hereditary properties of graphs

Mieczysław Borowiecki, Anna Fiedorowicz (2006)

Discussiones Mathematicae Graph Theory

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In this paper a concept 𝓠-Ramsey Class of graphs is introduced, where 𝓠 is a class of bipartite graphs. It is a generalization of well-known concept of Ramsey Class of graphs. Some 𝓠-Ramsey Classes of graphs are presented (Theorem 1 and 2). We proved that 𝓣₂, the class of all outerplanar graphs, is not 𝓓₁-Ramsey Class (Theorem 3). This results leads us to the concept of acyclic reducible bounds for a hereditary property 𝓟 . For 𝓣₂ we found two bounds (Theorem 4). An improvement,...

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 =...

The Thickness of Amalgamations and Cartesian Product of Graphs

Yan Yang, Yichao Chen (2017)

Discussiones Mathematicae Graph Theory

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The thickness of a graph is the minimum number of planar spanning subgraphs into which the graph can be decomposed. It is a measurement of the closeness to the planarity of a graph, and it also has important applications to VLSI design, but it has been known for only few graphs. We obtain the thickness of vertex-amalgamation and bar-amalgamation of graphs, the lower and upper bounds for the thickness of edge-amalgamation and 2-vertex-amalgamation of graphs, respectively. We also study...

Radio Graceful Hamming Graphs

Amanda Niedzialomski (2016)

Discussiones Mathematicae Graph Theory

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For k ∈ ℤ+ and G a simple, connected graph, a k-radio labeling f : V (G) → ℤ+ of G requires all pairs of distinct vertices u and v to satisfy |f(u) − f(v)| ≥ k + 1 − d(u, v). We consider k-radio labelings of G when k = diam(G). In this setting, f is injective; if f is also surjective onto {1, 2, . . . , |V (G)|}, then f is a consecutive radio labeling. Graphs that can be labeled with such a labeling are called radio graceful. In this paper, we give two results on the existence of radio...

The crossing numbers of join products of paths with graphs of order four

Marián Klešč, Stefan Schrötter (2011)

Discussiones Mathematicae Graph Theory

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Kulli and Muddebihal [V.R. Kulli, M.H. Muddebihal, Characterization of join graphs with crossing number zero, Far East J. Appl. Math. 5 (2001) 87-97] gave the characterization of all pairs of graphs which join product is planar graph. The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. There are only few results concerning crossing numbers of graphs obtained as join product of two graphs. In the paper, the exact values of crossing...

Graphs of low chordality.

Chandran, L.Sunil, Lozin, Vadim V., Subramanian, C.R. (2005)

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

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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.

Placing bipartite graphs of small size II

Beata Orchel (1996)

Discussiones Mathematicae Graph Theory

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In this paper we give all pairs of non mutually placeable (p,q)-bipartite graphs G and H such that 2 ≤ p ≤ q, e(H) ≤ p and e(G)+e(H) ≤ 2p+q-1.

On θ-graphs of partial cubes

Sandi Klavžar, Matjaz Kovse (2007)

Discussiones Mathematicae Graph Theory

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The Θ-graph Θ(G) of a partial cube G is the intersection graph of the equivalence classes of the Djoković-Winkler relation. Θ-graphs that are 2-connected, trees, or complete graphs are characterized. In particular, Θ(G) is complete if and only if G can be obtained from K₁ by a sequence of (newly introduced) dense expansions. Θ-graphs are also compared with familiar concepts of crossing graphs and τ-graphs.

A Characterization for 2-Self-Centered Graphs

Mohammad Hadi Shekarriz, Madjid Mirzavaziri, Kamyar Mirzavaziri (2018)

Discussiones Mathematicae Graph Theory

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A graph is called 2-self-centered if its diameter and radius both equal to 2. In this paper, we begin characterizing these graphs by characterizing edge-maximal 2-self-centered graphs via their complements. Then we split characterizing edge-minimal 2-self-centered graphs into two cases. First, we characterize edge-minimal 2-self-centered graphs without triangles by introducing specialized bi-independent covering (SBIC) and a structure named generalized complete bipartite graph (GCBG)....