Double domination critical and stable graphs upon vertex removal

Soufiane Khelifi; Mustapha Chellali

Displaying similar documents to “Double domination critical and stable graphs upon vertex removal”

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, $N_{G} (u) \cap D \neq N_{G} (v) \cap D$ . The locating-total domination number $γ_{L}^{t} (G)$ is the minimum cardinality of a locating-total...

On vertex stability with regard to complete bipartite subgraphs

Aneta Dudek, Andrzej Żak (2010)

Discussiones Mathematicae Graph Theory

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A graph G is called (H;k)-vertex stable if G contains a subgraph isomorphic to H ever after removing any of its k vertices. Q(H;k) denotes the minimum size among the sizes of all (H;k)-vertex stable graphs. In this paper we complete the characterization of $(K_{m, n}; 1)$ -vertex stable graphs with minimum size. Namely, we prove that for m ≥ 2 and n ≥ m+2, $Q (K_{m, n}; 1) = m n + m + n$ and $K_{m, n} * K ₁$ as well as $K_{m + 1, n + 1} - e$ are the only $(K_{m, n}; 1)$ -vertex stable graphs with minimum size, confirming the conjecture of Dudek and Zwonek.

Domination Subdivision Numbers

Teresa W. Haynes, Sandra M. Hedetniemi, Stephen T. Hedetniemi, David P. Jacobs, James Knisely, Lucas C. van der Merwe (2001)

Discussiones Mathematicae Graph Theory

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A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of V-S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number $s d_{γ} (G)$ 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 domination number. Arumugam conjectured that $1 \leq s d_{γ} (G) \leq 3$ for any graph G. We give a counterexample to this conjecture. On the other hand,...

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 $G$ ...

Graphs with large double domination numbers

Michael A. Henning (2005)

Discussiones Mathematicae Graph Theory

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In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number $γ_{\times 2} (G)$ . If G ≠ C₅ is a connected graph of order n with minimum degree at least 2, then we show that $γ_{\times 2} (G) \leq 3 n / 4$ and we characterize those graphs achieving equality.

A note on periodicity of the 2-distance operator

Bohdan Zelinka (2000)

Discussiones Mathematicae Graph Theory

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The paper solves one problem by E. Prisner concerning the 2-distance operator T₂. This is an operator on the class $C_{f}$ of all finite undirected graphs. If G is a graph from $C_{f}$ , then T₂(G) is the graph with the same vertex set as G in which two vertices are adjacent if and only if their distance in G is 2. E. Prisner asks whether the periodicity ≥ 3 is possible for T₂. In this paper an affirmative answer is given. A result concerning the periodicity 2 is added.

Weak roman domination in graphs

P. Roushini Leely Pushpam, T.N.M. Malini Mai (2011)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph and f be a function f:V → 0,1,2. A vertex u with f(u) = 0 is said to be undefended with respect to f, if it is not adjacent to a vertex with positive weight. The function f is a weak Roman dominating function (WRDF) if each vertex u with f(u) = 0 is adjacent to a vertex v with f(v) > 0 such that the function f’: V → 0,1,2 defined by f’(u) = 1, f’(v) = f(v)-1 and f’(w) = f(w) if w ∈ V-u,v, has no undefended vertex. The weight of f is $w (f) = \sum_{v \in V} f (v)$ . The weak Roman domination...

On arbitrarily vertex decomposable unicyclic graphs with dominating cycle

Sylwia Cichacz, Irmina A. Zioło (2006)

Discussiones Mathematicae Graph Theory

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A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n₁,...,nₖ) of positive integers such that $\sum_{i = 1}^{k} n_{i} = n$ , there exists a partition (V₁,...,Vₖ) of vertex set of G such that for every i ∈ 1,...,k the set $V_{i}$ induces a connected subgraph of G on $n_{i}$ vertices. We consider arbitrarily vertex decomposable unicyclic graphs with dominating cycle. We also characterize all such graphs with at most four hanging vertices such that exactly two of them have a common neighbour. ...

A remark on the (2,2)-domination number

Torsten Korneffel, Dirk Meierling, Lutz Volkmann (2008)

Discussiones Mathematicae Graph Theory

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A subset D of the vertex set of a graph G is a (k,p)-dominating set if every vertex v ∈ V(G)∖D is within distance k to at least p vertices in D. The parameter $γ_{k, p} (G)$ denotes the minimum cardinality of a (k,p)-dominating set of G. In 1994, Bean, Henning and Swart posed the conjecture that $γ_{k, p} (G) \leq (p / (p + k)) n (G)$ for any graph G with δₖ(G) ≥ k+p-1, where the latter means that every vertex is within distance k to at least k+p-1 vertices other than itself. In 2005, Fischermann and Volkmann confirmed this conjecture...

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...

(H,k) stable bipartite graphs with minimum size

Aneta Dudek, Małgorzata Zwonek (2009)

Discussiones Mathematicae Graph Theory

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Let us call a graph G(H;k) vertex stable if it contains a subgraph H after removing any of its k vertices. In this paper we are interested in finding the $(K_{n, n + 1}; 1)$ (respectively $(K_{n, n}; 1)$ ) vertex stable graphs with minimum size.

On locating and differentiating-total domination in trees

Mustapha Chellali (2008)

Discussiones Mathematicae Graph Theory

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A total dominating set of a graph G = (V,E) with no isolated vertex is a set S ⊆ V such that every vertex is adjacent to a vertex in S. A total dominating set S of a graph G is a locating-total dominating set if for every pair of distinct vertices u and v in V-S, N(u)∩S ≠ N(v)∩S, and S is a differentiating-total dominating set if for every pair of distinct vertices u and v in V, N[u]∩S ≠ N[v] ∩S. Let $γ ₜ^{L} (G)$ and $γ ₜ^{D} (G)$ be the minimum cardinality of a locating-total dominating set and a differentiating-total...

Signed total domination number of a graph

Bohdan Zelinka (2001)

Czechoslovak Mathematical Journal

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The signed total domination number of a graph is a certain variant of the domination number. If $v$ is a vertex of a graph $G$ , then $N (v)$ is its oper neighbourhood, i.e. the set of all vertices adjacent to $v$ in $G$ . A mapping $f : V (G) \to {- 1, 1}$ , where $V (G)$ is the vertex set of $G$ , is called a signed total dominating function (STDF) on $G$ , if $\sum_{x \in N (v)} f (x) \geq 1$ for each $v \in V (G)$ . The minimum of values $\sum_{x \in V (G)} f (x)$ , taken over all STDF’s of $G$ , is called the signed total domination number of $G$ and denoted by $γ_{s t} (G)$ . A theorem stating lower bounds for $γ_{s t} (G)$ is...

Domination in partitioned graphs

Zsolt Tuza, Preben Dahl Vestergaard (2002)

Discussiones Mathematicae Graph Theory

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Let V₁, V₂ be a partition of the vertex set in a graph G, and let $γ_{i}$ denote the least number of vertices needed in G to dominate $V_{i}$ . We prove that γ₁+γ₂ ≤ [4/5]|V(G)| for any graph without isolated vertices or edges, and that equality occurs precisely if G consists of disjoint 5-paths and edges between their centers. We also give upper and lower bounds on γ₁+γ₂ for graphs with minimum valency δ, and conjecture that γ₁+γ₂ ≤ [4/(δ+3)]|V(G)| for δ ≤ 5. As δ gets large, however, the largest...