The (a,b,s,t)-diameter of graphs: a particular case of conditional diameter

The conditional diameter of a connected graph $\Gamma=(V,E)$ is defined as follows: given a property ${\cal P}$ of a pair $(\Gamma_1, \Gamma_2)$ of subgraphs of $\Gamma$, the so-called \emph{conditional diameter} or ${\cal P}$-{\em diameter} measures the maximum distance among subgraphs satisfying ${\cal P}$. That is, $$D_{{\cal P}}(\Gamma):=\max_{\Gamma_1, \Gamma_2\subset \Gamma} \{\partial(\Gamma_1, \Gamma_2): \Gamma_1, \Gamma_2 \quad {\rm satisfy }\quad {\cal P}\}.$$ In this paper we consider the conditional diameter in which ${\cal P}$ requires that $\delta(u)\ge \alpha$ for all $u\in V(\Gamma_1)$, $\delta(v)\ge \beta$ for all $v\in V(\Gamma_2)$, $| V(\Gamma_1)| \ge s$ and $| V(\Gamma_2)| \ge t$ for some integers $1\le s,t\le |V|$ and $\delta \le \alpha, \beta \le \Delta$, where $\delta(x)$ denotes the degree of a vertex $x$ of $\Gamma$, $\delta$ denotes the minimum degree and $\Delta$ the maximum degree of $\Gamma$. The conditional diameter obtained is called $(\alpha ,\beta, s,t)$-\emph{diameter}. We obtain upper bounds on the $(\alpha ,\beta, s,t)$-diameter by using the $k$-alternating polynomials on the mesh of eigenvalues of an associated weighted graph. The method provides also bounds for other parameters such as vertex separators

Enumeration of s-d separators in DAGs with application to reliability analysis in temporal graphs

Temporal graphs are graphs in which arcs have temporal labels, specifying at which time they can be traversed. Motivated by recent results concerning the reliability analysis of a temporal graph through the enumeration of minimal cutsets in the corresponding line graph, in this paper we attack the problem of enumerating minimal s-d separators in s-d directed acyclic graphs (in short, s-d DAGs), also known as 2-terminal DAGs or s-t digraphs. Our main result is an algorithm for enumerating all the minimal s-d separators in a DAG with O(nm) delay, where n and m are respectively the number of nodes and arcs, and the delay is the time between the output of two consecutive solutions. To this aim, we give a characterization of the minimal s-d separators in a DAG through vertex cuts of an expanded version of the DAG itself. As a consequence of our main result, we provide an algorithm for enumerating all the minimal s-d cutsets in a temporal graph with delay O(m3), where m is the number of temporal arcs

Separability and Vertex Ordering of Graphs

Many graph optimization problems, such as finding an optimal coloring, or a largest clique, can be solved by a divide-and-conquer approach. One such well-known technique is decomposition by clique separators where a graph is decomposed into special induced subgraphs along their clique separators. While the most common practice of this method employs minimal clique separators, in this work we study other variations as well. We strive to characterize their structure and in particular the bound on the number of atoms. In fact, we strengthen the known bounds for the general clique cutset decomposition and the minimal clique separator decomposition. Graph ordering is the arrangement of a graph’s vertices according to a certain logic and is a useful tool in optimization problems. Special types of vertices are often recognized in graph classes, for instance it is well-known every chordal graph contains a simplicial vertex. Vertex-ordering, based on such properties, have originated many linear time algorithms. We propose to define a new family named SE-Class such that every graph belonging to this family inherently contains a simplicial extreme, that is a vertex which is either simplicial or has exactly two neighbors which are non-adjacent. Our family lends itself to an ordering based on simplicial extreme vertices (named SEO) which we demonstrate to be advantageous for the coloring and maximum clique problems. In addition, we examine the relation of SE-Class to the family of (Even-Hole, Kite)-free graphs and show a linear time generation of SEO for (Even-Hole, Diamond, Claw)-free graphs. We showcase the applications of those two core tools, namely clique-based decomposition and vertex ordering, on the (Even-Hole, Kite)-free family