742 research outputs found
Dominating sequences in grid-like and toroidal graphs
A longest sequence of distinct vertices of a graph such that each
vertex of dominates some vertex that is not dominated by its preceding
vertices, is called a Grundy dominating sequence; the length of is the
Grundy domination number of . In this paper we study the Grundy domination
number in the four standard graph products: the Cartesian, the lexicographic,
the direct, and the strong product. For each of the products we present a lower
bound for the Grundy domination number which turns out to be exact for the
lexicographic product and is conjectured to be exact for the strong product. In
most of the cases exact Grundy domination numbers are determined for products
of paths and/or cycles.Comment: 17 pages 3 figure
Identifying codes in vertex-transitive graphs and strongly regular graphs
We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2ln(vertical bar V vertical bar) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order vertical bar V vertical bar(alpha) with alpha is an element of{1/4, 1/3, 2/5}These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs
On the Roman domination in the lexicographic product of graphs
AbstractA Roman dominating function of a graph G=(V,E) is a function f:V→{0,1,2} such that every vertex with f(v)=0 is adjacent to some vertex with f(v)=2. The Roman domination number of G is the minimum of w(f)=∑v∈Vf(v) over all such functions. Using a new concept of the so-called dominating couple we establish the Roman domination number of the lexicographic product of graphs. We also characterize Roman graphs among the lexicographic product of graphs
- …