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Edge Roman domination on graphs
An edge Roman dominating function of a graph is a function satisfying the condition that every edge with
is adjacent to some edge with . The edge Roman
domination number of , denoted by , is the minimum weight
of an edge Roman dominating function of .
This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi
and Sadeghian Sadeghabad stating that if is a graph of maximum degree
on vertices, then . While the counterexamples having the edge Roman domination numbers
, we prove that is an upper bound for connected graphs. Furthermore, we
provide an upper bound for the edge Roman domination number of -degenerate
graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi
and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic
graphs.
In addition, we prove that the edge Roman domination numbers of planar graphs
on vertices is at most , which confirms a conjecture of
Akbari and Qajar. We also show an upper bound for graphs of girth at least five
that is 2-cell embeddable in surfaces of small genus. Finally, we prove an
upper bound for graphs that do not contain as a subdivision, which
generalizes a result of Akbari and Qajar on outerplanar graphs
Lower Bounds on the Distance Domination Number of a Graph
For an integer , a (distance) -dominating set of a connected graph is a set of vertices of such that every vertex of is at distance at most~ from some vertex of . The -domination number, , of is the minimum cardinality of a -dominating set of . In this paper, we establish lower bounds on the -domination number of a graph in terms of its diameter, radius, and girth. We prove that for connected graphs and , , where denotes the direct product of and
Upper bounds on the k-forcing number of a graph
Given a simple undirected graph and a positive integer , the
-forcing number of , denoted , is the minimum number of vertices
that need to be initially colored so that all vertices eventually become
colored during the discrete dynamical process described by the following rule.
Starting from an initial set of colored vertices and stopping when all vertices
are colored: if a colored vertex has at most non-colored neighbors, then
each of its non-colored neighbors becomes colored. When , this is
equivalent to the zero forcing number, usually denoted with , a recently
introduced invariant that gives an upper bound on the maximum nullity of a
graph. In this paper, we give several upper bounds on the -forcing number.
Notable among these, we show that if is a graph with order and
maximum degree , then . This simplifies to, for the zero forcing number case
of , . Moreover, when and the graph is -connected, we prove that , which is an improvement when , and
specializes to, for the zero forcing number case, . These results resolve a problem posed by
Meyer about regular bipartite circulant graphs. Finally, we present a
relationship between the -forcing number and the connected -domination
number. As a corollary, we find that the sum of the zero forcing number and
connected domination number is at most the order for connected graphs.Comment: 15 pages, 0 figure
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