5 research outputs found
Graphs where every k-subset of vertices is an identifying set
CombinatoricsInternational audienceLet be an undirected graph without loops and multiple edges. A subset is called \emph{identifying} if for every vertex the intersection of and the closed neighbourhood of is nonempty, and these intersections are different for different vertices . Let be a positive integer. We will consider graphs where \emph{every} -subset is identifying. We prove that for every the maximal order of such a graph is at most Constructions attaining the maximal order are given for infinitely many values of The corresponding problem of -subsets identifying any at most vertices is considered as well
Graphs where every k-subset of vertices is an identifying set
Let be an undirected graph without loops and multiple edges. A
subset is called \emph{identifying} if for every vertex
the intersection of and the closed neighbourhood of is nonempty, and
these intersections are different for different vertices .
Let be a positive integer. We will consider graphs where \emph{every}
-subset is identifying. We prove that for every the maximal order of
such a graph is at most Constructions attaining the maximal order are
given for infinitely many values of
The corresponding problem of -subsets identifying any at most
vertices is considered as well.Comment: 21 page
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
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 2 ln(|V|)+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 |V|^a with a in {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