11,351 research outputs found
Outer-convex domination in the corona of graphs
Let G be a connected simple graph. A subset S of a vertex set V (G) is called an outer-convex dominating set of G if for every vertex v ∈ V (G)\S, there exists a vertex x ∈ S such that xv is an edge of G and V (G)\S is a convex set. The outer-convex domination number of G, denoted by γecon(G), is the minimum cardinality of an outerconvex dominating set of G. In this paper, we show that every integers a, b, c, and n with a ≤ b ≤ c ≤ n − 1 is realizable as domination number, outer-connected domination number, outer-convex domination number, and order of G respectively. Further, we give the characterization of the outer-convex dominating set in the corona of two graphs and give its corresponding outer-convex domination number.Publisher's Versio
Further results on outer independent -rainbow dominating functions of graphs
Let be a graph. A function is a -rainbow dominating function if for every vertex
with , f\big{(}N(v)\big{)}=\{1,2\}. An outer-independent
-rainbow dominating function (OIRD function) of is a -rainbow
dominating function for which the set of all with
is independent. The outer independent -rainbow domination
number (OIRD number) is the minimum weight of an OIRD
function of .
In this paper, we first prove that is a lower bound on the OIRD
number of a connected claw-free graph of order and characterize all such
graphs for which the equality holds, solving an open problem given in an
earlier paper. In addition, a study of this parameter for some graph products
is carried out. In particular, we give a closed (resp. an exact) formula for
the OIRD number of rooted (resp. corona) product graphs and prove upper
bounds on this parameter for the Cartesian product and direct product of two
graphs
Domination Cover Pebbling: Structural Results
This paper continues the results of "Domination Cover Pebbling: Graph
Families." An almost sharp bound for the domination cover pebbling (DCP) number
for graphs G with specified diameter has been computed. For graphs of diameter
two, a bound for the ratio between the cover pebbling number of G and the DCP
number of G has been computed. A variant of domination cover pebbling, called
subversion DCP is introducted, and preliminary results are discussed.Comment: 15 page
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