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Displaying similar documents to “Connected partition dimensions of graphs”

Perfect connected-dominant graphs

Igor Edmundovich Zverovich (2003)

Discussiones Mathematicae Graph Theory

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If D is a dominating set and the induced subgraph G(D) is connected, then D is a connected dominating set. The minimum size of a connected dominating set in G is called connected domination number γ c ( G ) of G. A graph G is called a perfect connected-dominant graph if γ ( H ) = γ c ( H ) for each connected induced subgraph H of G.We prove that a graph is a perfect connected-dominant graph if and only if it contains no induced path P₅ and induced cycle C₅.

Connected domatic number in planar graphs

Bert L. Hartnell, Douglas F. Rall (2001)

Czechoslovak Mathematical Journal

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A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G . The connected domatic number of G is the maximum number of pairwise disjoint, connected dominating sets in V ( G ) . We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic...

Connected global offensive k-alliances in graphs

Lutz Volkmann (2011)

Discussiones Mathematicae Graph Theory

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We consider finite graphs G with vertex set V(G). For a subset S ⊆ V(G), we define by G[S] the subgraph induced by S. By n(G) = |V(G) | and δ(G) we denote the order and the minimum degree of G, respectively. Let k be a positive integer. A subset S ⊆ V(G) is a connected global offensive k-alliance of the connected graph G, if G[S] is connected and |N(v) ∩ S | ≥ |N(v) -S | + k for every vertex v ∈ V(G) -S, where N(v) is the neighborhood of v. The connected global offensive k-alliance number...

Connected resolvability of graphs

Varaporn Saenpholphat, Ping Zhang (2003)

Czechoslovak Mathematical Journal

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For an ordered set W = { w 1 , w 2 , , w k } of vertices and a vertex v in a connected graph G , the representation of v with respect to W is the k -vector r ( v | W ) = ( d ( v , w 1 ) , d ( v , w 2 ) , , d ( v , w k ) ) , where d ( x , y ) represents the distance between the vertices x and y . The set W is a resolving set for G if distinct vertices of G have distinct representations with respect to W . A resolving set for G containing a minimum number of vertices is a basis for G . The dimension dim ( G ) is the number of vertices in a basis for G . A resolving set W of G is connected...

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.

On odd and semi-odd linear partitions of cubic graphs

Jean-Luc Fouquet, Henri Thuillier, Jean-Marie Vanherpe, Adam P. Wojda (2009)

Discussiones Mathematicae Graph Theory

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A linear forest is a graph whose connected components are chordless paths. A linear partition of a graph G is a partition of its edge set into linear forests and la(G) is the minimum number of linear forests in a linear partition. In this paper we consider linear partitions of cubic simple graphs for which it is well known that la(G) = 2. A linear partition L = ( L B , L R ) is said to be odd whenever each path of L B L R has odd length and semi-odd whenever each path of L B (or each path of L R ) has odd length. In...

Total outer-connected domination in trees

Joanna Cyman (2010)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph. Set D ⊆ V(G) is a total outer-connected dominating set of G if D is a total dominating set in G and G[V(G)-D] is connected. The total outer-connected domination number of G, denoted by γ t c ( G ) , is the smallest cardinality of a total outer-connected dominating set of G. We show that if T is a tree of order n, then γ t c ( T ) 2 n / 3 . Moreover, we constructively characterize the family of extremal trees T of order n achieving this lower bound.

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...

The contractible subgraph of 5 -connected graphs

Chengfu Qin, Xiaofeng Guo, Weihua Yang (2013)

Czechoslovak Mathematical Journal

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An edge e of a k -connected graph G is said to be k -removable if G - e is still k -connected. A subgraph H of a k -connected graph is said to be k -contractible if its contraction results still in a k -connected graph. A k -connected graph with neither removable edge nor contractible subgraph is said to be minor minimally k -connected. In this paper, we show that there is a contractible subgraph in a 5 -connected graph which contains a vertex who is not contained in any triangles. Hence, every vertex...

Graph domination in distance two

Gábor Bacsó, Attila Tálos, Zsolt Tuza (2005)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph, and k ≥ 1 an integer. A subgraph D is said to be k-dominating in G if every vertex of G-D is at distance at most k from some vertex of D. For a given class of graphs, Domₖ is the set of those graphs G in which every connected induced subgraph H has some k-dominating induced subgraph D ∈ which is also connected. In our notation, Dom coincides with Dom₁. In this paper we prove that D o m D o m u = D o m u holds for u = all connected graphs without induced P u (u ≥ 2). (In particular,...