Related papers: Pappus-Desargues digraph confrontation

Finding a smallest subgraph that is k-edge-connected, or augmenting a k-edge-connected graph with a smallest subset of given candidate edges to become (k+1)-edge-connected, are among the most fundamental Network Design problems. They are…

A $2$-distance-transitive graph is a vertex-transitive graph whose vertex stabilizer is transitive on both the first step and the second step neighborhoods. In this paper, we first answer a question of A. Devillers, M. Giudici, C. H. Li and…

Local complement is a graph operation formalized by Bouchet which replaces the neighborhood of a chosen vertex with its edge-complement. This operation induces an equivalence relation on graphs; determining the size of the resulting…

We provide new tradeoffs between approximation and running time for the decremental all-pairs shortest paths (APSP) problem. For undirected graphs with $m$ edges and $n$ nodes undergoing edge deletions, we provide four new approximate…

In this paper, we study the Minimum Weight Pairwise Distance Preservers (MWPDP) problem. Consider a positively weighted undirected/directed connected graph $G = (V, E, c)$ and a subset $P$ of pairs of vertices, also called demand pairs. A…

This work introduces and systematically studies a new convex cone of PCOP (pairwise copositive). We establish that this cone is dual to the cone of PCP (pairwise completely positive) and, critically, provides a complete characterization for…

The L-intersection graphs are the graphs that have a representation as intersection graphs of axis parallel shapes in the plane. A subfamily of these graphs are {L, |, --}-contact graphs which are the contact graphs of axis parallel L, |,…

In this note we revisit the dart graph and the squared dart digraph constructions and prove that they yield strongly connected digraphs when applied to connected graphs of minimum valence at least 3.

We introduce a new decomposition of a graphs into quasi-4-connected components, where we call a graph quasi-4-connected if it is 3-connected and it only has separations of order 3 that remove a single vertex. Moreover, we give a cubic time…

We present a new family of graphs with remarkable properties. They are obtained by connecting the points of a random walk when their distance is smaller than a given scale. Their degree (number of neighbors) does not depend on the graph's…

A graph is k-linked if any k disjoint vertex-pairs can be joined by k disjoint paths. We improve a lower bound on the linkedness of polytopes slightly, which results in exact values for the minimal linkedness of 7-, 10- and 13-dimensional…

For a graph $G$ and $a,b\in V(G)$, the shortest path reconfiguration graph of $G$ with respect to $a$ and $b$ is denoted by $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths between $a$ and $b$ in $G$. Two vertices…

Half-arc-transitive graphs are a fascinating topic which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct…

To a graph, Hausel and Proudfoot associate two complex manifolds, B and D, which behave, respectively like moduli of local systems on a Riemann surface, and moduli of Higgs bundles. For instance, B is a moduli space of microlocal sheaves,…

Retraction note: After posting the manuscript on arXiv, we were informed by Erik Jan van Leeuwen that both results were known and they appeared in his thesis[vL09]. A PTAS for MDS is at Theorem 6.3.21 on page 79 and A PTAS for MCDS is at…

Let $S \subset \mathbb{R}^2$ be a set of $n$ sites, where each $s \in S$ has an associated radius $r_s > 0$. The disk graph $D(S)$ is the undirected graph with vertex set $S$ and an undirected edge between two sites $s, t \in S$ if and only…

Evidence that the pseudogap (PG) in a near-optimally doped Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ sample destroys the BCS logarithmic pairing instability [1] raises again the question of the role of the PG in the high-temperature…

Let $G$ be a graph on $n$ vertices, $p$ the order of a longest path and $\kappa$ the connectivity of $G$. In 1989, Bauer, Broersma Li and Veldman proved that if $G$ is a 2-connected graph with $d(x)+d(y)+d(z)\ge n+\kappa$ for all triples…

Bidirected graphs are a generalisation of directed graphs that arises in the study of undirected graphs with perfect matchings. Menger's famous theorem - the minimum size of a set separating two vertex sets $X$ and $Y$ is the same as the…

We investigate connected posets $C$ of height one as retracts of finite posets $P$. We define two multigraphs: a multigraph $\mathfrak{F}(P)$ reflecting the network of so-called improper 4-crown bundles contained in the extremal points of…