4,756 research outputs found
The Signed Domination Number of Cartesian Products of Directed Cycles
Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function is called a signed dominating function (SDF) iffor each vertex v. The weight w(f ) of f is defined by. The signed domination number of a digraph D is gs(D) = min{w(f ) : f is an SDF of D}. Let Cmn denotes the Cartesian product of directed cycles of length m and n. In this paper, we determine the exact value of signed domination number of some classes of Cartesian product of directed cycles. In particular, we prove that: (a) gs(C3n) = n if n 0(mod 3), otherwise gs(C3n) = n + 2. (b) gs(C4n) = 2n. (c) gs(C5n) = 2n if n 0(mod 10), gs(C5n) = 2n + 1 if n 3, 5, 7(mod 10), gs(C5n) = 2n + 2 if n 2, 4, 6, 8(mod 10), gs(C5n) = 2n + 3 if n 1,9(mod 10). (d) gs(C6n) = 2n if n 0(mod 3), otherwise gs(C6n) = 2n + 4. (e) gs(C7n) = 3n
The Italian domination numbers of some products of directed cycles
An Italian dominating function on a digraph with vertex set is
defined as a function such that every vertex
with has at least two in-neighbors assigned under
or one in-neighbor with . In this paper, we determine the
exact values of the Italian domination numbers of some products of directed
cycles
Vertex decomposable graphs and obstructions to shellability
Inspired by several recent papers on the edge ideal of a graph G, we study
the equivalent notion of the independence complex of G. Using the tool of
vertex decomposability from geometric combinatorics, we show that 5-chordal
graphs with no chordless 4-cycles are shellable and sequentially
Cohen-Macaulay. We use this result to characterize the obstructions to
shellability in flag complexes, extending work of Billera, Myers, and Wachs. We
also show how vertex decomposability may be used to show that certain graph
constructions preserve shellability.Comment: 13 pages, 3 figures. v2: Improved exposition, added Section 5.2 and
additional references. v3: minor corrections for publicatio
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