Minimum 22-edge strongly biconnected spanning directed subgraph problem

Wu and Grumbach introduced the concept of strongly biconnected directed graphs. A directed graph $G=(V,E)$ is called strongly biconnected if the directed graph $G$ is strongly connected and the underlying undirected graph of $G$ is biconnected. A strongly biconnected directed graph $G=(V,E)$ is said to be $2$- edge strongly biconnected if it has at least three vertices and the directed subgraph $(V,E\setminus\left\lbrace e\right\rbrace )$ is strongly biconnected for all $e \in E$. Let $G=(V,E)$ be a $2$-edge-strongly biconnected directed graph. In this paper we study the problem of computing a minimum size subset $H \subseteq E$ such that the directed subgraph $(V,H)$ is $2$- edge strongly biconnected

Computing 22-twinless blocks

Let $G=(V,E))$ be a directed graph. A $2$-twinless block in $G$ is a maximal vertex set $B\subseteq V$ of size at least $2$ such that for each pair of distinct vertices $x,y \in B$, and for each vertex $w\in V\setminus\left\lbrace x,y \right\rbrace$, the vertices $x,y$ are in the same twinless strongly connected component of $G\setminus\left \lbrace w \right\rbrace$. In this paper we present algorithms for computing the $2$-twinless blocks of a directed graph

Minimum 22-vertex strongly biconnected spanning directed subgraph problem

A directed graph $G=(V,E)$ is strongly biconnected if $G$ is strongly connected and its underlying graph is biconnected. A strongly biconnected directed graph $G=(V,E)$ is called $2$-vertex-strongly biconnected if $|V|\geq 3$ and the induced subgraph on $V\setminus\left\lbrace w\right\rbrace$ is strongly biconnected for every vertex $w\in V$. In this paper we study the following problem. Given a $2$-vertex-strongly biconnected directed graph $G=(V,E)$, compute an edge subset $E^{2sb} \subseteq E$ of minimum size such that the subgraph $(V,E^{2sb})$ is $2$-vertex-strongly biconnected

22-edge-twinless blocks

Let $G=(V,E)$ be a directed graph. A $2$-edge-twinless block in $G$ is a maximal vertex set $C^{t}\subseteq V$ with $|C^{t}|>1$ such that for any distinct vertices $v,w \in C^{t}$, and for every edge $e\in E$, the vertices $v,w$ are in the same twinless strongly connected component of $G\setminus\left \lbrace e \right\rbrace$. In this paper we study this concept and describe algorithms for computing $2$-edge-twinless blocks

Computing the 22-blocks of directed graphs

Let $G$ be a directed graph. A \textit{$2$-directed block} in $G$ is a maximal vertex set $C^{2d}\subseteq V$ with $|C^{2d}|\geq 2$ such that for each pair of distinct vertices $x,y \in C^{2d}$, there exist two vertex-disjoint paths from $x$ to $y$ and two vertex-disjoint paths from $y$ to $x$ in $G$. In contrast to the $2$-vertex-connected components of $G$, the subgraphs induced by the $2$-directed blocks may consist of few or no edges. In this paper we present two algorithms for computing the $2$-directed blocks of $G$ in $O(\min\lbrace m,(t_{sap}+t_{sb})n\rbrace n)$ time, where $t_{sap}$ is the number of the strong articulation points of $G$ and $t_{sb}$ is the number of the strong bridges of $G$. Furthermore, we study two related concepts: the $2$-strong blocks and the $2$-edge blocks of $G$. We give two algorithms for computing the $2$-strong blocks of $G$ in $O( \min \lbrace m,t_{sap} n\rbrace n)$ time and we show that the $2$-edge blocks of $G$ can be computed in $O(\min \lbrace m, t_{sb} n \rbrace n)$ time. In this paper we also study some optimization problems related to the strong articulation points and the $2$-blocks of a directed graph. Given a strongly connected graph $G=(V,E)$, find a minimum cardinality set $E^{*}\subseteq E$ such that $G^{*}=(V,E^{*})$ is strongly connected and the strong articulation points of $G$ coincide with the strong articulation points of $G^{*}$. This problem is called minimum strongly connected spanning subgraph with the same strong articulation points. We show that there is a linear time $17/3$ approximation algorithm for this NP-hard problem. We also consider the problem of finding a minimum strongly connected spanning subgraph with the same $2$-blocks in a strongly connected graph $G$. We present approximation algorithms for three versions of this problem, depending on the type of $2$-blocks