Distance in stratified graphs

Gary Chartrand; Lisa Hansen; Reza Rashidi; Naveed Sherwani

Displaying similar documents to “Distance in stratified graphs”

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...

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.

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...

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...

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...

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 the minus domination number of graphs

Hailong Liu, Liang Sun (2004)

Czechoslovak Mathematical Journal

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Let $G = (V, E)$ be a simple graph. A $3$ -valued function $f V (G) \to {- 1, 0, 1}$ is said to be a minus dominating function if for every vertex $v \in V$ , $f (N [v]) = \sum_{u \in N [v]} f (u) \geq 1$ , where $N [v]$ is the closed neighborhood of $v$ . The weight of a minus dominating function $f$ on $G$ is $f (V) = \sum_{v \in V} f (v)$ . The minus domination number of a graph $G$ , denoted by $γ^{-} (G)$ , equals the minimum weight of a minus dominating function on $G$ . In this paper, the following two results are obtained. (1) If $G$ is a bipartite graph of order $n$ , then $γ^{-} (G) \geq 4 (\sqrt{n + 1} - 1) - n .$ (2) For any negative integer $k$ and any positive integer...

Homogeneously embedding stratified graphs in stratified graphs

Gary Chartrand, Donald W. Vanderjagt, Ping Zhang (2005)

Mathematica Bohemica

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A 2-stratified graph $G$ is a graph whose vertex set has been partitioned into two subsets, called the strata or color classes of $G$ . Two $2$ -stratified graphs $G$ and $H$ are isomorphic if there exists a color-preserving isomorphism $φ$ from $G$ to $H$ . A $2$ -stratified graph $G$ is said to be homogeneously embedded in a $2$ -stratified graph $H$ if for every vertex $x$ of $G$ and every vertex $y$ of $H$ , where $x$ and $y$ are colored the same, there exists an induced $2$ -stratified subgraph $H^{'}$ of $H$ containing $y$ and a color-preserving...