16,164 research outputs found
New bounds on the signed total domination number of graphs
In this paper, we study the signed total domination number in graphs and
present new sharp lower and upper bounds for this parameter. For example by
making use of the classic theorem of Turan, we present a sharp lower bound on
this parameter for graphs with no complete graph of order r+1 as a subgraph.
Also, we prove that n-2(s-s') is an upper bound on the signed total domination
number of any tree of order n with s support vertices and s' support vertives
of degree two. Moreover, we characterize all trees attainig this bound.Comment: This paper contains 11 pages and one figur
On the inverse signed total domination number in graphs
In this paper, we study the inverse signed total domination number in graphs
and present new lower and upper bounds on this parameter. For example by making
use of the classic theorem of Turan (1941), we present a sharp upper bound for
graphs with no induced complete subgraph of order greater than two. Also, we
bound this parameter for a tree in terms of its order and the number of leaves
and characterize all trees attaining this bound
Domination changing and unchanging signed graphs upon the vertex removal
A subset S of V (Σ) is a dominating set of Σ if |N⁺(v) ∩ S| > |N⁻(v) ∩ S| for all v ∈ V − S. This article is to start a study of those signed graphs that are stable and critical in the following way: If the removal of an arbitrary vertex does not change the domination number, the signed graph will be stable. The signed graph, on the other hand, is unstable if an arbitrary vertex is removed and the domination number changes. Specifically, we analyze the change in the domination of the vertex deletion and stable signed graphs.Publisher's Versio
Some Bounds on the Double Domination of Signed Generalized Petersen Graphs and Signed I-Graphs
In a graph , a vertex dominates itself and its neighbors. A subset is a double dominating set of if dominates every vertex
of at least twice. A signed graph is a graph
together with an assignment of positive or negative signs to all its
edges. A cycle in a signed graph is positive if the product of its edge signs
is positive. A signed graph is balanced if all its cycles are positive. A
subset is a double dominating set of if it
satisfies the following conditions: (i) is a double dominating set of ,
and (ii) is balanced, where
is the subgraph of induced by the edges of with one end point
in and the other end point in . The cardinality of a minimum
double dominating set of is the double domination number
. In this paper, we give bounds for the double
domination number of signed cubic graphs. We also obtain some bounds on the
double domination number of signed generalized Petersen graphs and signed
I-graphs.Comment: 13 page
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