Displaying similar documents to “Connected domination critical graphs with respect to relative complements”

A characterization of locating-total domination edge critical graphs

Mostafa Blidia, Widad Dali (2011)

Discussiones Mathematicae Graph Theory

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For a graph G = (V,E) without isolated vertices, a subset D of vertices of V is a total dominating set (TDS) of G if every vertex in V is adjacent to a vertex in D. The total domination number γₜ(G) is the minimum cardinality of a TDS of G. A subset D of V which is a total dominating set, is a locating-total dominating set, or just a LTDS of G, if for any two distinct vertices u and v of V(G)∖D, N G ( u ) ∩ D ≠ N G ( v ) ∩ D . The locating-total domination number γ L t ( G ) is the minimum cardinality of a locating-total...

Double domination critical and stable graphs upon vertex removal

Soufiane Khelifi, Mustapha Chellali (2012)

Discussiones Mathematicae Graph Theory

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In a graph a vertex is said to dominate itself and all its neighbors. A double dominating set of a graph G is a subset of vertices that dominates every vertex of G at least twice. The double domination number of G, denoted γ × 2 ( G ) , is the minimum cardinality among all double dominating sets of G. We consider the effects of vertex removal on the double domination number of a graph. A graph G is γ × 2 -vertex critical graph ( γ × 2 -vertex stable graph, respectively) if the removal of any vertex different...

Distance in stratified graphs

Gary Chartrand, Lisa Hansen, Reza Rashidi, Naveed Sherwani (2000)

Czechoslovak Mathematical Journal

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A graph G is stratified if its vertex set is partitioned into classes, called strata. If there are k strata, then G is k -stratified. These graphs were introduced to study problems in VLSI design. The strata in a stratified graph are also referred to as color classes. For a color X in a stratified graph G , the X -eccentricity e X ( v ) of a vertex v of G is the distance between v and an X -colored vertex furthest from v . The minimum X -eccentricity among the vertices of G is the X -radius r a d X G of G ...

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...

Inequalities involving independence domination, f -domination, connected and total f -domination numbers

San Ming Zhou (2000)

Czechoslovak Mathematical Journal

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Let f be an integer-valued function defined on the vertex set V ( G ) of a graph G . A subset D of V ( G ) is an f -dominating set if each vertex x outside D is adjacent to at least f ( x ) vertices in D . The minimum number of vertices in an f -dominating set is defined to be the f -domination number, denoted by γ f ( G ) . In a similar way one can define the connected and total f -domination numbers γ c , f ( G ) and γ t , f ( G ) . If f ( x ) = 1 for all vertices x , then these are the ordinary domination number, connected domination number and total...

Coalescing Fiedler and core vertices

Didar A. Ali, John Baptist Gauci, Irene Sciriha, Khidir R. Sharaf (2016)

Czechoslovak Mathematical Journal

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The nullity of a graph G is the multiplicity of zero as an eigenvalue in the spectrum of its adjacency matrix. From the interlacing theorem, derived from Cauchy’s inequalities for matrices, a vertex of a graph can be a core vertex if, on deleting the vertex, the nullity decreases, or a Fiedler vertex, otherwise. We adopt a graph theoretical approach to determine conditions required for the identification of a pair of prescribed types of root vertices of two graphs to form a cut-vertex...

Matchings and total domination subdivision number in graphs with few induced 4-cycles

Odile Favaron, Hossein Karami, Rana Khoeilar, Seyed Mahmoud Sheikholeslami (2010)

Discussiones Mathematicae Graph Theory

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A set S of vertices of a graph G = (V,E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γₜ(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number s d γ ₜ ( G ) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. Favaron, Karami, Khoeilar and Sheikholeslami (Journal...