The forcing convexity number of a graph

Gary Chartrand; Ping Zhang

Displaying similar documents to “The forcing convexity number of a graph”

$H$ -convex graphs

Gary Chartrand, Ping Zhang (2001)

Mathematica Bohemica

Similarity:

For two vertices $u$ and $v$ in a connected graph $G$ , the set $I (u, v)$ consists of all those vertices lying on a $u - v$ geodesic in $G$ . For a set $S$ of vertices of $G$ , the union of all sets $I (u, v)$ for $u, v \in S$ is denoted by $I (S)$ . A set $S$ is convex if $I (S) = S$ . The convexity number $c o n (G)$ is the maximum cardinality of a proper convex set in $G$ . A convex set $S$ is maximum if $| S | = c o n (G)$ . The cardinality of a maximum convex set in a graph $G$ is the convexity number of $G$ . For a nontrivial connected graph $H$ , a connected graph $G$ is an $H$ -convex graph...

Graphs with convex domination number close to their order

Joanna Cyman, Magdalena Lemańska, Joanna Raczek (2006)

Discussiones Mathematicae Graph Theory

Similarity:

For a connected graph G = (V,E), a set D ⊆ V(G) is a dominating set of G if every vertex in V(G)-D has at least one neighbour in D. The distance $d_{G} (u, v)$ between two vertices u and v is the length of a shortest (u-v) path in G. An (u-v) path of length $d_{G} (u, v)$ is called an (u-v)-geodesic. A set X ⊆ V(G) is convex in G if vertices from all (a-b)-geodesics belong to X for any two vertices a,b ∈ X. A set X is a convex dominating set if it is convex and dominating. The convex domination number $γ_{c o n} (G)$ of a...

The vertex detour hull number of a graph

A.P. Santhakumaran, S.V. Ullas Chandran (2012)

Discussiones Mathematicae Graph Theory

Similarity:

For vertices x and y in a connected graph G, the detour distance D(x,y) is the length of a longest x - y path in G. An x - y path of length D(x,y) is an x - y detour. The closed detour interval ID[x,y] consists of x,y, and all vertices lying on some x -y detour of G; while for S ⊆ V(G), $I_{D} [S] = ⋃_{x, y \in S} I_{D} [x, y]$ . A set S of vertices is a detour convex set if $I_{D} [S] = S$ . The detour convex hull ${[S]}_{D}$ is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among subsets S of...

Convex universal fixers

Magdalena Lemańska, Rita Zuazua (2012)

Discussiones Mathematicae Graph Theory

Similarity:

In [1] Burger and Mynhardt introduced the idea of universal fixers. Let G = (V, E) be a graph with n vertices and G’ a copy of G. For a bijective function π: V(G) → V(G’), define the prism πG of G as follows: V(πG) = V(G) ∪ V(G’) and $E (π G) = E (G) \cup E (G^{'}) \cup M_{π}$ , where $M_{π} = u π (u) | u \in V (G)$ . Let γ(G) be the domination number of G. If γ(πG) = γ(G) for any bijective function π, then G is called a universal fixer. In [9] it is conjectured that the only universal fixers are the edgeless graphs K̅ₙ. In this work we generalize the concept...

A note on periodicity of the 2-distance operator

Bohdan Zelinka (2000)

Discussiones Mathematicae Graph Theory

Similarity:

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.

On the minus domination number of graphs

Hailong Liu, Liang Sun (2004)

Czechoslovak Mathematical Journal

Similarity:

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...

Distance in stratified graphs

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

Czechoslovak Mathematical Journal

Similarity:

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 $G$ ...

On some results about convex functions of order $α$

M. Obradović, S. Owa (1986)

Matematički Vesnik

Similarity: