We investigate metric dimension and the localization game for several families of directed analogues of strongly regular graphs and their generalizations, adapting a probabilistic method of Babai (1980) for bounding the size of resolving sets in undirected strongly regular graphs. We derive upper bounds on the localization number and metric dimension depending on the order of the graph and the maximum number of common out-neighbours for a pair of vertices. We consider normally regular digraphs, so-called"ordinary graphs", classes of Deza digraphs, divisible design digraphs, nearly doubly regular tournaments, and certain doubly regular team tournaments. In particular, for asymmetric normally regular digraphs on $n$ vertices, we show that these invariants are bounded above by $O(\sqrt{n} \log n)$, and improve this to $O(\log n)$ for a class of doubly regular team tournaments.
A resolving set in a graph is a set of landmarks whose distance vectors distinguish all vertices. We use information theory to prove lower bounds for metric dimension and class dimension in distance-regular graphs and association schemes arising from finite geometry. The core idea is that, for a fixed landmark, a rando...
The main theorem gives the asymptotic sampling distribution and enumeration formulae for configurations, and accommodates forbidden edges, and enables the sampling of edge-colored graphs with prescribed degree sequences for each color class by constructing the colored subgraphs one at a time.
I. Kryven, Rik Versendaal, Mike de Vries· 0 citations
This work presents $O(n^2)$-time optimal algorithms for $n$-vertex graphs coming from a class of linear neighborhood complexity for the following problems: All-Pairs Shortest Paths, and the multiplication of the adjacency matrix of the input graph with any $n \times n$ matrix.
Édouard Bonnet, Julien Duron, M. Pilipczuk et al.· 0 citations
For a graph $G$, let $h(G)$ be the minimum cardinality of a vertex set meeting every maximum independent set of $G$. We establish two complementary reduction principles for the Bollob\'as--Erd\H{o}s--Tuza conjecture: the conjecture for arbitrary graphs is equivalent to its restriction to regular graphs of any fixed pos...
We introduce and study a new family of circulant digraphs associated with the cyclic group \({\mathbb Z}_N\), obtained by restricting admissible combinations of two generators \(a\) and \(b\) to three coordinate sectors. The resulting distance-like function differs from the standard directed distance in circulant digra...
C. Dalf'o, M. Fiol, M. Reyes· Utilitas mathematica· 0 citations
We study a restriction of the classical Erd\H{o}s--S\'os problem, the extremal number of trees, to the class of bipartite host graphs, both when only the order of the host is prescribed and when its two part-sizes are fixed. We give natural lower-bound constructions and formulate corresponding linear upper-bound conjec...
Luke Waite, Nuh Aydin· 0 citations
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