Bounding the Openk-Monopoly Number of Strong Product Graphs

Dorota Kuziak; Iztok Peterin; Ismael G. Yero

Displaying similar documents to “Bounding the Openk-Monopoly Number of Strong Product Graphs”

The hull number of strong product graphs

A.P. Santhakumaran, S.V. Ullas Chandran (2011)

Discussiones Mathematicae Graph Theory

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For a connected graph G with at least two vertices and S a subset of vertices, the convex hull ${[S]}_{G}$ is the smallest convex set containing S. The hull number h(G) is the minimum cardinality among the subsets S of V(G) with ${[S]}_{G} = V (G)$ . Upper bound for the hull number of strong product G ⊠ H of two graphs G and H is obtainted. Improved upper bounds are obtained for some class of strong product graphs. Exact values for the hull number of some special classes of strong product graphs are obtained. Graphs...

On the total restrained domination number of direct products of graphs

Wai Chee Shiu, Hong-Yu Chen, Xue-Gang Chen, Pak Kiu Sun (2012)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph. A total restrained dominating set is a set S ⊆ V where every vertex in V∖S is adjacent to a vertex in S as well as to another vertex in V∖S, and every vertex in S is adjacent to another vertex in S. The total restrained domination number of G, denoted by $γ_{r}^{t} (G)$ , is the smallest cardinality of a total restrained dominating set of G. We determine lower and upper bounds on the total restrained domination number of the direct product of two graphs. Also, we show that...

On well-covered graphs of odd girth 7 or greater

Bert Randerath, Preben Dahl Vestergaard (2002)

Discussiones Mathematicae Graph Theory

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A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [14] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. One of the most challenging problems in this area, posed in the survey of Plummer [15], is to find a good characterization of well-covered graphs of girth 4. We examine several subclasses of well-covered graphs of girth ≥ 4 with respect to the odd girth...

Edge-connectivity of strong products of graphs

Bostjan Bresar, Simon Spacapan (2007)

Discussiones Mathematicae Graph Theory

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The strong product G₁ ⊠ G₂ of graphs G₁ and G₂ is the graph with V(G₁)×V(G₂) as the vertex set, and two distinct vertices (x₁,x₂) and (y₁,y₂) are adjacent whenever for each i ∈ 1,2 either $x_{i} = y_{i}$ or $x_{i} y_{i} \in E (G_{i})$ . In this note we show that for two connected graphs G₁ and G₂ the edge-connectivity λ (G₁ ⊠ G₂) equals minδ(G₁ ⊠ G₂), λ(G₁)(|V(G₂)| + 2|E(G₂)|), λ(G₂)(|V(G₁)| + 2|E(G₁)|). In addition, we fully describe the structure of possible minimum edge cut sets in strong products of graphs.

A Finite Characterization and Recognition of Intersection Graphs of Hypergraphs with Rank at Most 3 and Multiplicity at Most 2 in the Class of Threshold Graphs

Yury Metelsky, Kseniya Schemeleva, Frank Werner (2017)

Discussiones Mathematicae Graph Theory

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We characterize the class [...] L32 $L_{3}^{2}$ of intersection graphs of hypergraphs with rank at most 3 and multiplicity at most 2 by means of a finite list of forbidden induced subgraphs in the class of threshold graphs. We also give an O(n)-time algorithm for the recognition of graphs from [...] L32 $L_{3}^{2}$ in the class of threshold graphs, where n is the number of vertices of a tested graph.

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...

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...

Radio numbers for generalized prism graphs

Paul Martinez, Juan Ortiz, Maggy Tomova, Cindy Wyels (2011)

Discussiones Mathematicae Graph Theory

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A radio labeling is an assignment c:V(G) → N such that every distinct pair of vertices u,v satisfies the inequality d(u,v) + |c(u)-c(v)| ≥ diam(G) + 1. The span of a radio labeling is the maximum value. The radio number of G, rn(G), is the minimum span over all radio labelings of G. Generalized prism graphs, denoted $Z_{n, s}$ , s ≥ 1, n ≥ s, have vertex set (i,j) | i = 1,2 and j = 1,...,n and edge set ((i,j),(i,j ±1)) ∪ ((1,i),(2,i+σ)) | σ = -⌊(s-1)/2⌋...,0,...,⌊s/2⌋. In this paper we determine...

Factorizations of properties of graphs

Izak Broere, Samuel John Teboho Moagi, Peter Mihók, Roman Vasky (1999)

Discussiones Mathematicae Graph Theory

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A property of graphs is any isomorphism closed class of simple graphs. For given properties of graphs ₁,₂,...,ₙ a vertex (₁, ₂, ...,ₙ)-partition of a graph G is a partition V₁,V₂,...,Vₙ of V(G) such that for each i = 1,2,...,n the induced subgraph $G [V_{i}]$ has property $_{i}$ . The class of all graphs having a vertex (₁, ₂, ...,ₙ)-partition is denoted by ₁∘₂∘...∘ₙ. A property is said to be reducible with respect to a lattice of properties of graphs if there are n ≥ 2 properties ₁,₂,...,ₙ ∈ such that...

Rotation and jump distances between graphs

Gary Chartrand, Heather Gavlas, Héctor Hevia, Mark A. Johnson (1997)

Discussiones Mathematicae Graph Theory

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A graph H is obtained from a graph G by an edge rotation if G contains three distinct vertices u,v, and w such that uv ∈ E(G), uw ∉ E(G), and H = G-uv+uw. A graph H is obtained from a graph G by an edge jump if G contains four distinct vertices u,v,w, and x such that uv ∈ E(G), wx∉ E(G), and H = G-uv+wx. If a graph H is obtained from a graph G by a sequence of edge jumps, then G is said to be j-transformed into H. It is shown that for every two graphs G and H of the same order (at least...

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),...

The order of uniquely partitionable graphs

Izak Broere, Marietjie Frick, Peter Mihók (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 V₁,...,Vₙ of V(G) such that, for each i = 1,...,n, the subgraph of G induced by $V_{i}$ has property $_{i}$ . If a graph G has a unique (₁,...,ₙ)-partition we say it is uniquely (₁,...,ₙ)-partitionable. We establish best lower bounds for the order of uniquely (₁,...,ₙ)-partitionable graphs, for various choices of ₁,...,ₙ.

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...

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.

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.

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.

New edge neighborhood graphs

Ali A. Ali, Salar Y. Alsardary (1997)

Czechoslovak Mathematical Journal

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Let $G$ be an undirected simple connected graph, and $e = u v$ be an edge of $G$ . Let $N_{G} (e)$ be the subgraph of $G$ induced by the set of all vertices of $G$ which are not incident to $e$ but are adjacent to $u$ or $v$ . Let $𝒩_{e}$ be the class of all graphs $H$ such that, for some graph $G$ , $N_{G} (e) ≅ H$ for every edge $e$ of $G$ . Zelinka [3] studied edge neighborhood graphs and obtained some special graphs in $𝒩_{e}$ . Balasubramanian and Alsardary [1] obtained some other graphs in $𝒩_{e}$ . In this paper we given some new graphs in $𝒩_{e}$ .

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 =...

Radio k-labelings for Cartesian products of graphs

Mustapha Kchikech, Riadh Khennoufa, Olivier Togni (2008)

Discussiones Mathematicae Graph Theory

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Frequency planning consists in allocating frequencies to the transmitters of a cellular network so as to ensure that no pair of transmitters interfere. We study the problem of reducing interference by modeling this by a radio k-labeling problem on graphs: For a graph G and an integer k ≥ 1, a radio k-labeling of G is an assignment f of non negative integers to the vertices of G such that $| f (x) - f (y) | \geq k + 1 - d_{G} (x, y)$ , for any two vertices x and y, where $d_{G} (x, y)$ is the distance between x and y in G. The radio k-chromatic...

Total domination of Cartesian products of graphs

Xinmin Hou (2007)

Discussiones Mathematicae Graph Theory

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Let γₜ(G) and $γ_{p r} (G)$ denote the total domination and the paired domination numbers of graph G, respectively, and let G □ H denote the Cartesian product of graphs G and H. In this paper, we show that γₜ(G)γₜ(H) ≤ 5γₜ(G □ H), which improves the known result γₜ(G)γₜ(H) ≤ 6γₜ(G □ H) given by Henning and Rall.

Fall coloring of graphs I

Rangaswami Balakrishnan, T. Kavaskar (2010)

Discussiones Mathematicae Graph Theory

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A fall coloring of a graph G is a proper coloring of the vertex set of G such that every vertex of G is a color dominating vertex in G (that is, it has at least one neighbor in each of the other color classes). The fall coloring number $χ_{f} (G)$ of G is the minimum size of a fall color partition of G (when it exists). Trivially, for any graph G, $χ (G) \leq χ_{f} (G)$ . In this paper, we show the existence of an infinite family of graphs G with prescribed values for χ(G) and $χ_{f} (G)$ . We also obtain the smallest non-fall...