A characterization of roman trees

Michael A. Henning

Displaying similar documents to “A characterization of roman trees”

On well-covered graphs of odd girth 7 or greater

Bert Randerath, Preben Dahl Vestergaard (2002)

Discussiones Mathematicae Graph Theory

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A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [14] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. One of the most challenging problems in this area, posed in the survey of Plummer [15], is to find a good characterization of well-covered graphs of girth 4. We examine several subclasses of well-covered graphs of girth ≥ 4 with respect to the odd girth...

On integral sum graphs with a saturated vertex

Zhibo Chen (2010)

Czechoslovak Mathematical Journal

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As introduced by F. Harary in 1994, a graph $G$ is said to be an $i n t e g r a l$ $s u m$ $g r a p h$ if its vertices can be given a labeling $f$ with distinct integers so that for any two distinct vertices $u$ and $v$ of $G$ , $u v$ is an edge of $G$ if and only if $f (u) + f (v) = f (w)$ for some vertex $w$ in $G$ . We prove that every integral sum graph with a saturated vertex, except the complete graph $K_{3}$ , has edge-chromatic number equal to its maximum degree. (A vertex of a graph $G$ is said to be if it is adjacent to every...

Minus total domination in graphs

Hua Ming Xing, Hai-Long Liu (2009)

Czechoslovak Mathematical Journal

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A three-valued function $f V \to {- 1, 0, 1}$ defined on the vertices of a graph $G = (V, E)$ is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. That is, for every $v \in V$ , $f (N (v)) \geq 1$ , where $N (v)$ consists of every vertex adjacent to $v$ . The weight of an MTDF is $f (V) = \sum f (v)$ , over all vertices $v \in V$ . The minus total domination number of a graph $G$ , denoted $γ_{t}^{-} (G)$ , equals the minimum weight of an MTDF of $G$ . In this paper, we discuss some properties of minus total domination on a graph...

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.

The cobondage number of a graph

V.R. Kulli, B. Janakiram (1996)

Discussiones Mathematicae Graph Theory

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A set D of vertices in a graph G = (V,E) is a dominating set of G if every vertex in V-D is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set. We define the cobondage number $b_{c} (G)$ of G to be the minimum cardinality among the sets of edges X ⊆ P₂(V) - E, where P₂(V) = X ⊆ V:|X| = 2 such that γ(G+X) < γ(G). In this paper, the exact values of bc(G) for some standard graphs are found and some bounds are obtained. Also, a Nordhaus-Gaddum...

Uniquely partitionable planar graphs with respect to properties having a forbidden tree

Jozef Bucko, Jaroslav Ivančo (1999)

Discussiones Mathematicae Graph Theory

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Let ₁, ₂ be graph properties. A vertex (₁,₂)-partition of a graph G is a partition V₁,V₂ of V(G) such that for i = 1,2 the induced subgraph $G [V_{i}]$ has the property $_{i}$ . A property ℜ = ₁∘₂ is defined to be the set of all graphs having a vertex (₁,₂)-partition. A graph G ∈ ₁∘₂ is said to be uniquely (₁,₂)-partitionable if G has exactly one vertex (₁,₂)-partition. In this note, we show the existence of uniquely partitionable planar graphs with respect to hereditary additive properties having a...

On total restrained domination in graphs

De-xiang Ma, Xue-Gang Chen, Liang Sun (2005)

Czechoslovak Mathematical Journal

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In this paper we initiate the study of total restrained domination in graphs. Let $G = (V, E)$ be a graph. A total restrained dominating set is a set $S \subseteq V$ where every vertex in $V - S$ is adjacent to a vertex in $S$ as well as to another vertex in $V - S$ , and every vertex in $S$ is adjacent to another vertex in $S$ . The total restrained domination number of $G$ , denoted by $γ_{r}^{t} (G)$ , is the smallest cardinality of a total restrained dominating set of $G$ . First, some exact values and sharp bounds for $γ_{r}^{t} (G)$ are given in Section 2....