A directed graph $G=(V,E)$ is called strongly biconnected if $G$ is strongly connected and the underlying graph of $G$ is biconnected. A strongly biconnected component of a strongly connected graph $G=(V,E)$ is a maximal vertex subset $L\subseteq V$ such that the induced subgraph on $L$ is strongly biconnected. Let $G=(V,E)$ be a strongly biconnected directed graph. A $2$-edge-biconnected block in $G$ is a maximal vertex subset $U\subseteq V$ such that for any two distict vertices $v,w \in U$ and for each edge $b\in E$, the vertices $v,w$ are in the same strongly biconnected components of $G\setminus\left\lbrace b\right\rbrace $. A $2$-strong-biconnected block in $G$ is a maximal vertex subset $U\subseteq V$ of size at least $2$ such that for every pair of distinct vertices $v,w\in U$ and for every vertex $z\in V\setminus\left\lbrace v,w \right\rbrace $, the vertices $v$ and $w$ are in the same strongly biconnected component of $G\setminus \left\lbrace v,w \right\rbrace $. In this paper we study $2$-edge-biconnected blocks and $2$-strong biconnected blocks.
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…
Let $G=(V,E)$ be a strongly biconnected directed graph. In this paper we consider the problem of computing an edge subset $H \subseteq E$ of minimum size such that the directed subgraph $(V,H)$ is strongly biconnected.
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.
A directed graph $G=(V,E)$ is called strongly biconnected if $G$ is strongly connected and the underlying graph of $G$ is biconnected. This class of directed graphs was first introduced by Wu and Grumbach. Let $G=(V,E)$ be a strongly biconnected directed graph. An edge $e\in E$ is a b-bridge if the subgraph $G\setminus \left\lbrace e\right\rbrace =(V,E\setminus \left\lbrace e\right\rbrace) $ is not strongly biconnected. A vertex $w\in V$ is a b-articulation point if $G\setminus \left\lbrace w\right\rbrace$ is not strongly biconnected, where $G\setminus \left\lbrace w\right\rbrace$ is the subgr…
A strongly connected graph is strongly biconnected if after ignoring the direction of its edges we have an undirected graph with no articulation points. A 3-vertex strongly biconnected graph is a strongly biconnected digraph that has the property that deleting any two vertices in this graph leaves a strongly binconnected subgraph. Jaberi [11] presented approximation algorithms for minimum cardinality 2-vertex strongly biconnected directed subgraph problem. We will focus in this paper on polynomial time algorithms which we have implemented for producing spanning subgraphs that are 3-vertex stro…
The SPQR-tree is a data structure that compactly represents all planar embeddings of a biconnected planar graph. It plays a key role in constrained planarity testing. We develop a similar data structure, called the UP-tree, that compactly represents all upward planar embeddings of a biconnected single-source directed graph. We demonstrate the usefulness of the UP-tree by solving the upward planar embedding extension problem for biconnected single-source directed graphs.
Jaberi [7] presented approximation algorithms for the problem of computing a minimum size 2-vertex strongly biconnected subgraph in directed graphs. We have implemented approximation algorithms presented in [7] and we have tested the implementation on some graphs. The experimental results show that these algorithms work well in practice.
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.
A directed graph $G=(V,E)$ is twinless strongly connected if it contains a strongly connected spanning subgraph without any pair of antiparallel (or twin) edges. The twinless strongly connected components (TSCCs) of a directed graph $G$ are its maximal twinless strongly connected subgraphs. These concepts have several diverse applications, such as the design of telecommunication networks and the structural stability of buildings. A vertex $v \in V$ is a twinless strong articulation point of $G$ if the deletion of $v$ increases the number of TSCCs of $G$. Here, we present the first linear-time …
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.
The eternal vertex cover problem is a dynamic variant of the classical vertex cover problem. It is NP-hard to compute the eternal vertex cover number of graphs and known algorithmic results for the problem are very few. This paper presents a linear time recursive algorithm for computing the eternal vertex cover number of cactus graphs. Unlike other graph classes for which polynomial time algorithms for eternal vertex cover number are based on efficient computability of a known lower bound directly derived from minimum vertex cover, we show that it is a certain substructure property that helps …
A directed acyclic hypergraph is a generalized concept of a directed acyclic graph, where each hyperedge can contain an arbitrary number of tails and heads. Directed hypergraphs can be used to model data flow and execution dependencies in streaming applications. Thus, hypergraph partitioning algorithms can be used to obtain efficient parallelizations for multiprocessor architectures. However, an acyclicity constraint on the partition is necessary when mapping streaming applications to embedded multiprocessors due to resource restrictions on this type of hardware. The acyclic hypergraph partiti…
