2 -blocks in strongly biconnected directed graphs
aa r X i v : . [ c s . D S ] J u l Raed Jaberi
Abstract
A directed graph G = ( V, E ) is called strongly biconnected if G is stronglyconnected and the underlying graph of G is biconnected. A strongly bicon-nected component of a strongly connected graph G = ( V, E ) is a maximalvertex subset L ⊆ V such that the induced subgraph on L is strongly bi-connected. Let G = ( V, E ) be a strongly biconnected directed graph. A2-edge-biconnected block in G is a maximal vertex subset U ⊆ V such thatfor any two distict vertices v, w ∈ U and for each edge b ∈ E , the vertices v, w are in the same strongly biconnected components of G \ { b } . A 2-strong-biconnected block in G is a maximal vertex subset U ⊆ V of size at least2 such that for every pair of distinct vertices v, w ∈ U and for every vertex z ∈ V \ { v, w } , the vertices v and w are in the same strongly biconnectedcomponent of G \ { v, w } . In this paper we study 2-edge-biconnected blocksand 2-strong biconnected blocks. Keywords:
Directed graphs, Graph algorithms, Strongly biconnecteddirected graphs, 2-blocks
1. Introduction
Let G = ( V, E ) be a directed graph. A 2-edge block of G is a maximalvertex subset L e ⊆ V with | L e | > x, y ∈ L e , G contains two edge-disjoint paths from x to y and twoedge-disjoint paths from y to x . A 2-strong block of G is a maximal vertexsubset B s ⊆ V with | B s | > x, y ∈ B s and for every vertex w ∈ V \ { x, y } , x and y belong to the samestrongly connected component of G \ { w } . G is called strongly biconnectedif G is strongly connected and the underlying graph of G is biconnected.This class of directed graphs was introduced by Wu and Grumbach [11]. Astrongly biconnected component of G is a maximal vertex subset C ⊆ V suchthat the induced subgraph on C is strongly biconnected [11]. Let G = ( V, E )be a strongly biconnected directed graph. An edge e ∈ E is a b-bridge ifthe subgraph G \ { e } = ( V, E \ { e } ) is not strongly biconnected. A vertex Preprint submitted to arXiv July 21, 2020 ∈ V is a b-articulation point if G \ { w } is not strongly biconnected, where G \ { w } is the subgraph obtained from G by deleting w . G is 2-edge-strongly-biconnected (respectively, 2-vertex-strongly biconnected) if | V | > G has no b-bridges (respectively, b-articulation points). A 2-edge-biconnectedblock in G is a maximal vertex subset U ⊆ V such that for any two distictvertices v, w ∈ U and for each edge b ∈ E , the vertices v, w are in thesame strongly biconnected components of G \ { b } . The 2-edge blocks of G are disjoint [5]. Notice that 2-edge-biconnected blocks are not necessarilydisjoint, as shown in Figure 1. 1 23 4 5 67 89 1011 121314 1516 Figure 1: A strongly biconnected directed graph G . The vertex subset { , , , , , , , } is a 2-edge block of G . Notice that the vertices 15 ,
12 are notin the same 2-edge-biconnected block because 15 and 12 are not in the same stronglybiconnected component of G \ { (2 , } . Moreover, G has two 2-edge-biconnected blocks U = { , , , , , , } and U = { , } . U and U share vertex 4 A 2-strong-biconnected block in G is a maximal vertex subset U ⊆ V ofsize at least 2 such that for every pair of distinct vertices v, w ∈ U and forevery vertex z ∈ V \ { v, w } , the vertices v and w are in the same stronglybiconnected component of G \ { z } . 2locks, articulation points, and bridges of an undirected graph can be cal-culated in O ( n + m ) time [9, 10, 8]. In[4], Georgiadis presented a linear timealgorithm to test whether a directed graph is 2-vertex-connected. Strong ar-ticulation points and strong bridges of a directed graph can be computed in O ( n + m ) time [7, 1]. Jaberi [5] presented algorithms for computing 2-strongblocks, and 2-edge blocks of a directed graph. Georgiadis et al. [2, 3] gavelinear time algorithms for determining 2-edge blocks and 2-strong blocks. Wuand Grumbach [11] introduced the concept of strongly biconnected directedgraphs and the concept of strongly biconnected components. Jaberi [6] stud-ied b-bridges in strongly biconnected directed graphs. In this paper we study2-edge-biconnected blocks and 2-strong biconnected blocks.
2. 2-edge-biconnected blocks
In this section we study 2-edge-biconnected blocks and present an algo-rithm for computing them. Let G = ( V, E ) be a strongly biconnected directedgraph. For every pair of distinct vertices x, y ∈ V , we write x e ! y if for anyedge b ∈ E , the vertices x, y belong to the same strongly biconnected com-ponent of G \ { b } . A 2-edge-biconnected blocks in G is a maximal subset ofvertices U ⊆ V with | U | > x, y ∈ U , we have x e ! y . A 2-edge-strongly-biconnected component in G is a maximal vertexsubset C eb ⊆ V such that the induced subgraph on C eb is 2-edge-stronglybiconnected. Note that the strongly biconnected directed graph in Figure1 contains one 2-edge-strongly biconnected component { , , , , , } ,which is a subset of the 2-edge-biconnected block { , , , , , , } . Lemma 2.1.
Let G = ( V, E ) be a strongly biconnected directed graph and let C eb be a -edge-strongly biconnected component of G . Then C eb is a subsetof a -edge-biconnected block of G .Proof. Let x and y be distinct vertices in C eb and let e ∈ E . Let G [ C eb ]be the induced subgraph on C eb . By definition, the subgraph obtained from G [ C eb ] by deleting e is still strongly biconnected. Therefore, we have x e ! y . (cid:3) Lemma 2.2.
Let U , U be distinct -edge-biconnected blocks of a stronglybiconnected directed graph G = ( V, E ) . Then | U ∩ U | ≤ roof. Assume for the purpose of contradiction that | U ∩ U | >
1. Let x ∈ U \ ( U ∩ U ) and let y ∈ U \ ( U ∩ U ). Let v, w ∈ U ∩ U with v = w and let b ∈ E . Notice that x, v belong to the same stongly connectedcomponent of G \ { b } since x e ! v . Moreover, v, y belong to the same stonglyconnected component of G \ { b } . Consequently, x, y are in the same stonglyconnected component of G \ { b } . Then, the vertices x, y do not lie in thesame strongly biconnected component of G \ { b } . Suppose that C x , C y aretwo strongly biconnected components of G \{ b } such that x ∈ C x and y ∈ C y .There are two cases to consider.1. v ∈ C x ∩ C y . In this case w / ∈ C x ∩ C y . Suppose without loss of generalitythat w ∈ C x . Then w, y are not in the same stongly biconnectedcomponents of G \ { b } . But this contradicts that w e ! y v / ∈ C x ∩ C y . Suppose without loss of generality that v ∈ C x . Then v, y do not lie in the same strongly biconnected component of G \ { b } . Butthis contradicts that v e ! y (cid:3) Using similar arguments as in Lemma 2.2, we can prove the following.
Lemma 2.3.
Let G = ( V, E ) be a strongly biconnected directed graph and let { w , w , . . . , w t } ⊆ V such that w e ! w t and w i − e ! w i for i ∈ { , . . . , t } .Then { w , w , . . . , w t } is a subset of a -edge biconnected block of G . Lemma 2.4.
Let G = ( V, E ) be a strongly biconnected directed graph andlet v, w be two distinct vertices in G . Let b be an edge in G such that b isnot a b-bridge. Then, the vertices v, w are in the same strongly biconnectedcomponent of G \ { b } Proof. immediate from definition. (cid:3)
Algorithm 2.5 shows an algorithm for computing all the 2-edge bicon-nected blocks of a strongly biconnected directed graph.The correctness of this algorithm follows from Lemma 2.2, Lemma 2.3,and Lemma 2.4.
Theorem 2.6.
Algorithm 2.5 runs in O ( n ) time.Proof. The b-bridges of G can be computed in O ( nm ) time [6]. Stronglybiconnected components can be calculated in linear time [11]. Lines 7–11take O ( b.n ), where b is the number of b-bridges in G . The time required forbuilding G eb in lines 12–16 is O ( n ). Moreover, the blocks of an undirectedgraph can be found in linear time using Tarjan’s algorithm. [9, 8].. (cid:3) lgorithm 2.5.Input: A strongly biconnected directed graph G = ( V, E ). Output:
The 2-edge biconnected blocks of G . Compute the b-bridges of G If G has no b-bridges then . Output V . else Let L be an n × n matrix. Initialize L with 1s. for each b–bridge b of G do calculate the strongly biconnected components of G \ { b } for each pair ( x, y ) ∈ V × V do if x, y in different strongly biconnected components of G \ { b } then L [ x, y ] ← E eb ← ∅ . for every pair ( x, y ) ∈ V × V do if L [ x, y ] = 1 and L [ y, x ] = 1 then E eb ← E eb ∪ { ( x, y ) } G eb ← ( V, E eb ) Compute all the blocks of size ≥ G eb and output them.
3. 2-strong-biconnected blocks
In this section we illustrate some properties of 2-strong-biconnected blocks.The strongly biconnected directed graph in Figure 2 has two 2-strong bicon-nected blocks L = { , , , } and L = { , . . } . Note that L and L share two vertices. The intersection of any two distinct 2-strong biconnectedblocks contains at most 2 vertices. Note also that the subgraph induced bythe 2-strong biconnected block L has no edges.Let G = ( V, E ) be a strongly biconnected directed graph. A 2-vertex-strongly biconnected component C sb is a maximal vertex subset C sb ⊆ V such that the induced subgraph on C sb is 2-vertex-strongly biconnected.Each 2-vertex-strongly biconnected component C sb of G is a subset of a2-strong-biconnected-block of G . Furthermore, each 2-vertex-strongly bicon-nected component C sb of G is 2-vertex connected. Therefore, the subgraphinduced by C sb contains at least 2 | C sb | edges. In contrast to, the subgraphsinduced by the 2-strong-biconnected blocks do not necessarily contain edges.5 23 45 678910111213 14 15 1617 18 19 20 Figure 2: A strongly biconnected directed graph G = ( V, E ). The vertices 2 and 6 are inthe same 2-strong block of G but they do not belong to the same 2-strong-biconnected-block of G since they are not in the same strongly biconnected component of G \ { } .Moreover, G has two 2-strong biconnected blocks L = { , , , } and L = { , . . } .Note that | L ∩ L | = 2 eferenceseferences