Further topics in connectivity

Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called “conditional connectivity,” are considered. For unexplained terminology concerning connectivity, see §4.1.Peer ReviewedPostprint (published version

Maximally Edge-Connected Realizations and Kundu's kk-factor Theorem

A simple graph $G$ with edge-connectivity $\lambda(G)$ and minimum degree $\delta(G)$ is maximally edge connected if $\lambda(G)=\delta(G)$. In 1964, given a non-increasing degree sequence $\pi=(d_{1},\ldots,d_{n})$, Jack Edmonds showed that there is a realization $G$ of $\pi$ that is $k$-edge-connected if and only if $d_{n}\geq k$ with $\sum_{i=1}^{n}d_{i}\geq 2(n-1)$ when $d_{n}=1$. We strengthen Edmonds's result by showing that given a realization $G_{0}$ of $\pi$ if $Z_{0}$ is a spanning subgraph of $G_{0}$ with $\delta(Z_{0})\geq 1$ such that $|E(Z_{0})|\geq n-1$ when $\delta(G_{0})=1$, then there is a maximally edge-connected realization of $\pi$ with $G_{0}-E(Z_{0})$ as a subgraph. Our theorem tells us that there is a maximally edge-connected realization of $\pi$ that differs from $G_{0}$ by at most $n-1$ edges. For $\delta(G_{0})\geq 2$, if $G_{0}$ has a spanning forest with $c$ components, then our theorem says there is a maximally edge-connected realization that differs from $G_{0}$ by at most $n-c$ edges. As an application we combine our work with Kundu's $k$-factor Theorem to find maximally edge-connected realizations with a $(k_{1},\dots,k_{n})$-factor for $k\leq k_{i}\leq k+1$ and present a partial result to a conjecture that strengthens the regular case of Kundu's $k$-factor theorem.Comment: 13 pages, 1 figur

Graph Theory

This was a workshop on graph theory, with a comprehensive approach. Highlights included the emerging theories of sparse graph limits and of infinite matroids, new techniques for colouring graphs on surfaces, and extensions of graph minor theory to directed graphs and to immersion

Extremal Directed And Mixed Graphs

We consider three problems in extremal graph theory, namely the degree/diameter problem, the degree/geodecity problem and Tur\'{a}n problems, in the context of directed and partially directed graphs. A directed graph or mixed graph $G$ is $k$-geodetic if there is no pair of vertices $u,v$ of $G$ such that there exist distinct non-backtracking walks with length $\leq k$ in $G$ from $u$ to $v$. The order of a $k$-geodetic digraph with minimum out-degree $d$ is bounded below by the \emph{directed Moore bound} $M(d,k) = 1+d+d^2+\dots +d^k$; similarly the order of a $k$-geodetic mixed graph with minimum undirected degree $r$ and minimum directed out-degree $z$ is bounded below by the \emph{mixed Moore bound}. We will be interested in networks with order exceeding the Moore bound by some small \emph{excess} $\epsilon$. The \emph{degree/geodecity problem} asks for the smallest possible order of a $k$-geodetic digraph or mixed graph with given degree parameters. We prove the existence of extremal graphs, which we call \emph{geodetic cages}, and provide some bounds on their order and information on their structure. We discuss the structure of digraphs with excess one and rule out the existence of certain digraphs with excess one. We then classify all digraphs with out-degree two and excess two, as well as all diregular digraphs with out-degree two and excess three. We also present the first known non-trivial examples of geodetic cages. We then generalise this work to the setting of mixed graphs. First we address the question of the total regularity of mixed graphs with order close to the Moore bound and prove bounds on the order of mixed graphs that are not totally regular. In particular using spectral methods we prove a conjecture of L\'{o}pez and Miret that mixed graphs with diameter two and order one less than the Moore bound are totally regular. Using counting arguments we then provide strong bounds on the order of totally regular $k$-geodetic mixed graphs and use these results to derive new extremal mixed graphs. Finally we change our focus and study the Tur\'{a}n problem of the largest possible size of a $k$-geodetic digraph with given order. We solve this problem and also prove an exact expression for the restricted problem of the largest possible size of strongly connected $2$-geodetic digraphs, as well as providing constructions of strongly connected $k$-geodetic digraphs that we conjecture to be extremal for larger $k$. We close with a discussion of some related generalised Tur\'{a}n problems for $k$-geodetic digraphs

Proceedings of the 3rd International Workshop on Optimal Networks Topologies IWONT 2010

Peer Reviewe

Link Patterns in Complex Networks

Network theorists define patterns in complex networks in various ways to make them accessible to human beholders. Prominent definitions are thereby based on the partition of the network's nodes into groups such that underlying patterns in the link structure become apparent. Clustering and blockmodeling are two well-known approaches of this kind. In this thesis, we treat pattern search problems as discrete mathematical optimization problems. From this viewpoint, we develop a new mathematical classification of clustering and blockmodeling approaches, which unifies these two fields and replaces several NP-hardness proofs by a single one. We furthermore use this classification to develop integer mathematical programming formulations for pattern search problems and discuss new linearization techniques for polynomial functions therein. We apply these results to a model for a new pattern search problem. Even though it is the most basic problem in combinatorial terms, we can prove its NP-hardness. In fact, we show that it is a generalization of well-known problems including the Traveling Salesman and the Quadratic Assignment Problem. Our derived exact pattern search procedure is up to 10,000 times faster than comparable methods from the literature. To demonstrate its practicability, we finally apply the procedure to the world trade network from the United Nations' database and show that the network deviates by less than 0.14% from the patterns we found

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum