Upper minus domination in regular graphs

2007

J. of Shanghai Univ.

Self Cite

For an arbitrary subset P of the reals, a function f : V → P is defined to be a P-dominating function of a graph G = (V, E) if the sum of its function values over any closed neighbourhood is at least 1. That is, for everyThe definition of total P-dominating function is obtained by simply changing 'closed' neighborhood N [v] in the definition of P-dominating function to 'open' neighborhood N (v). The (total) P-domination number of a graph G is defined to be the infimum of weight w(f ) = P v∈V f (v) taken over all (total) P-dominating function f . Similarly, the P-edge and P-star dominating functions can be defined. In this paper we survey some recent progress on the topic of dominating functions in graph theory. Especially, we are interested in P-, P-edge and P-star dominating functions of graphs with integer values.

Section: Minus Domination In Graphsmentioning

confidence: 94%

“…In [20] the authors asked for the upper bounds on Γ − for a cubic graph. In [52] this problem was answered by giving a sharp upper bound on Γ − for a cubic graph. In 2000, Kang and Cai [53] further generalized the result to k-regular graphs as follows.…”

Section: Minus Domination In Graphsmentioning

confidence: 99%

Dominating functions with integer values in graphs—a survey

Kang

2007

J. of Shanghai Univ.

Self Cite

“…First, we obtain n = (q + m) + p = (q + m) + (p 00 + p 01 + p 10 + p 20 ) + (p 11 + p 30 ) (q + m) + p 11 + (p 11 + p 30 ) + (q 10 + 2q 21 ) + (m 13 + 2m 21 ) (by (14)) (9), (10)),…”

Section: The Equality Holds If and Only If G ∈ Fmentioning

confidence: 98%

Upper minus total domination in small-degree regular graphs

Yan

Yang

2007

Discrete Mathematics

A function f : V (G) → {−1, 0, 1} defined on the vertices of a graph G is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. An MTDF f is minimal if there does not exist an MTDF g:The weight of an MTDF is the sum of its function values over all vertices. The minus total domination number of G is the minimum weight of an MTDF on G, while the upper minus domination number of G is the maximum weight of a minimal MTDF on G. In this paper we present upper bounds on the upper minus total domination number of a cubic graph and a 4-regular graph and characterize the regular graphs attaining these upper bounds.

“…In [12], Kang et al further extended the result to k-partite graphs for k ≥ 2. Other results and progress on the research for signed and minus domination of graphs can be found in [1,2,4,10,11,14,16,17,19].…”

mentioning

confidence: 99%

An application of the Turán theorem to domination in graphs

Discrete Applied Mathematics

Cheng

Kang

2008

A function f : V (G) → {+1, −1} defined on the vertices of a graph G is a signed dominating function if for any vertex v the sum of function values over its closed neighborhood is at least one. The signed domination number γ s (G) of G is the minimum weight of a signed dominating function on G. By simply changing "{+1, −1}" in the above definition to "{+1, 0, −1}", we can define the minus dominating function and the minus domination number of G. In this note by applying Turán theorem we present sharp lower bounds on the signed domination number for a graph containing no (k + 1)-cliques. As a result, we generalize a previous result due to Kang et al. on the minus domination number of k-partite graphs to graphs containing no (k + 1)-cliques and characterize the extremal graphs.