Abstract
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 $|V|^{\alpha}$ with $\alpha \in \{\frac{1}{4},\frac{1}{3},\frac{2}{5}\}$. These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs.
Highlights
Given a discrete structure on a set of elements, a natural question is to be able to locate efficiently the elements using the structure
Babai [1] gave an upper bound on the size of the symmetric differences of open neighbourhood in strongly regular graphs which leads to bounds on the metric dimension
We provide identifying codes for several vertex-transitive families of graphs which have size of the same order as the fractional value
Summary
Given a discrete structure on a set of elements, a natural question is to be able to locate efficiently the elements using the structure. Identifying codes have already been studied in different classes of vertex-transitive graphs, especially in cycles [6, 21, 28, 39] and hypercubes [7, 12, 13, 27, 29] In these examples, the order of the size of an optimal identifying code seems to always match its fractional value. Paley graphs give an example of an infinite family of graphs for which the optimal value of fractional identifying code is constant but the integer value is logarithmic, and so the gap between the two is logarithmic We consider another family of strongly regular graphs that have never been studied in the context of identifying codes nor resolving sets: the adjacency graphs of generalized quadrangles.
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