On arbitrarily vertex decomposable unicyclic graphs with dominating cycle

Sylwia Cichacz; Irmina A. Zioło

Displaying similar documents to “On arbitrarily vertex decomposable unicyclic graphs with dominating cycle”

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...

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 $γ_{\times 2} (G)$ , 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 $γ_{\times 2}$ -vertex critical graph ( $γ_{\times 2}$ -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 $G$ ...

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...

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...

Pairs of forbidden class of subgraphs concerning $K_{1, 3}$ and P₆ to have a cycle containing specified vertices

Takeshi Sugiyama, Masao Tsugaki (2009)

Discussiones Mathematicae Graph Theory

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In [3], Faudree and Gould showed that if a 2-connected graph contains no $K_{1, 3}$ and P₆ as an induced subgraph, then the graph is hamiltonian. In this paper, we consider the extension of this result to cycles passing through specified vertices. We define the families of graphs which are extension of the forbidden pair $K_{1, 3}$ and P₆, and prove that the forbidden families implies the existence of cycles passing through specified vertices.

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,...

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...

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.

Graceful signed graphs

Mukti Acharya, Tarkeshwar Singh (2004)

Czechoslovak Mathematical Journal

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A $(p, q)$ -sigraph $S$ is an ordered pair $(G, s)$ where $G = (V, E)$ is a $(p, q)$ -graph and $s$ is a function which assigns to each edge of $G$ a positive or a negative sign. Let the sets $E^{+}$ and $E^{-}$ consist of $m$ positive and $n$ negative edges of $G$ , respectively, where $m + n = q$ . Given positive integers $k$ and $d$ , $S$ is said to be $(k, d)$ -graceful if the vertices of $G$ can be labeled with distinct integers from the set ${0, 1, \dots, k + (q - 1) d}$ such that when each edge $u v$ of $G$ is assigned the product of its sign and the absolute difference of the integers assigned to...

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.

Restrained domination in unicyclic graphs

Johannes H. Hattingh, Ernst J. Joubert, Marc Loizeaux, Andrew R. Plummer, Lucas van der Merwe (2009)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S. The restrained domination number of G, denoted by $γ_{r} (G)$ , is the minimum cardinality of a restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We show that if U is a unicyclic graph of order n, then $γ_{r} (U) \geq ⎡ n / 3 ⎤$ , and provide a characterization of graphs achieving this bound.

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.

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...