May 1987
UILU-ENG-8 -2232
ACT-77
COORDINATED SCIENCE LABORATORY
College of Engineering
ON FINDING
THE VERTEX
CONNECTIVITY
OF GRAPHS
Milind Girkar
Milind Sohoni
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
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11. TITLE (Include Security Classification)
On Finding the Vertex Connectivity of Graphs
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Girkar, Milind and Sohoni, Milind
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May 1987
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Vertex connectivity, .Graph algorithm, Network flows
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An implementation of the fastest known algorithm to find the vertex connectivity of
graphs with reduced space requirement is presented.
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On finding the vertex connectivity of graphs1
Milind Girkar2
Milind Sohoni3
Abstract
A n implementation o f the fastest known algorithm to find the vertex connectivity o f graphs
with reduced space requirement is presented.
1. In trodu ction
Let G [V ,E ) be a finite undirected graph with no self-loops and no parallel edges. A set of
vertices, S, is called an (a ,6) vertex separator if { a ^ C V - S and every path connecting a and b
passes through at least one vertex o f S. Clearly if a and b are connected by an edge, no (a ,6)
vertex separator exists. W e define N G(a,b) to be \V\-1 if (a,b)eE, else it is the least cardinality o f
an (a ,b) vertex separator. The vertex connectivity o f G, kG is defined to be mina le v N a (a}b).
When kG is small, there are well-known linear time algorithms to determine connectivity
(*g>0)> biconnectivity ( ^ > 1 ) (see e.g., [4]) and triconnectivity (kG>2) [8,11]. There is an o (| v f)
algorithm [9] to check four-connectivity (fcG>3); others [3,5,7] are o f 0(|V|[S|). For a fixed it,
there are some randomized algorithms [1,10] for testing it-connectivity.
In this paper we consider the question o f determining kG1 when ka is large. For this prob­
lem, the only known deterministic methods to find it depend on solving maximum flow problems
in unit networks [5,7]. (A unit network has the property that the capacity o f each edge is one
1 This work was supported in part by the Joint Services Electronics Program under Grant No. N 00014-84-C -0149.
* Center for Supercomputing Research and Development, University o f Illinois at Urbana-Champaign, Urbana, IL 61801. The
work o f this author was supported in part by the National Science Foundation under Grants No. NSF D CR84-10110 and NSF
D CR84-06918, the U.S. Department o f Energy under Grant No. DOE DE-FG02-85ER25001 and the IBM Donation.
1 Department o f Computer Science, University o f Illinois at Urbana-Champaign, Urbana, IL 61801.
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and every vertex other than the source or sink has either only one edge emanating from it or one
edge entering it.) O f these, the most efficient one is G alil’s [7] with a running time o f
0 (max(A:(?,|V’|V4)t 0 \EHvf4) with a space requirement o f
0 [[k G+\V\)\E\).
^mProve uPon this
result by presenting an algorithm that has the same running time as G alil’s but with a space
requirement o f only 0(|V||^|).
2. E ven ’s A lgorith m
In [3] Even solves the simpler problem (denoted by P G k) o f finding whether kG> k , for a
given G and k. Even’s algorithm is as follows:
Let V = {v l,v2, • • •
following way.
and let Lj=*{vv vv • • •
Define G• to be the graph constructed in the
Gj contains all the vertices and edges o f G; in addition it includes a new vertex t
connected by an edge to each vertex in L-.
(1)
For every i and j such that l < » < / < * , check whether iVc (v,.,t>y)>ifc. If for some i and j this
test fails then halt; kG< k .
(2)
G.
For every j such that k+l<j<\V\, form G} and check whether N }[s,v .)> k . If for some ;
this test fails then halt; kG< k .
(3)
Halt; kG> k .
Whether N {a,b)> k can be found out by checking that the value o f the maximum flow in the
corresponding network is at least k. This involves finding k flow augmenting paths (f.a.p.’s) in
the network using the Ford and Fulkerson [6] algorithm. A f.a.p. can be found in 0(|2?D time
and since k f.a .p .’s need to be found in at most k2+\V\ flow problems, the complexity o f Even’s
algorithm is 0 {k 3\E[+k\V\\E\).
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In [7] Galil observes that Even’s algorithm can be used to find kG by progressively solving
^ g,v ? g,2> ’ ' ’ u^til P Gtk+l yields a negative answer; then kG= k. By using D inic’s algorithm [2] to
find augmenting paths and modifying Even’s algorithm, Galil shows that this can be done in
0(majc(fcG,|V'f6)£G|V'f6[Z?D using
|VDl2?D memory. Using an approach similar to Galil’s we
get a reduced space bound.
3. The Algorithm
First we simplify Even’s algorithm as follows:
In the first step instead o f checking whether N ° (vitvj)>k, we do some additional work and
find N {vitv.) and then trivially check whether this is greater than or equal to k. It will turn out
that the extra work will not change the time complexity o f the algorithm.
The outline o f the algorithm is as follows.
(1)
Initialize Jfc to 1, MIN to \V[-l.
(2)
For every %such that l < t < £ - l , find N G(v.,vk).
(3)
Use the results o f step 2 to update MIN to min(min1<J<4_1JV<?(t;.,wJk),MrJV)
(4)
If M IN<k then halt; kG=MIN.
(5)
For every ; such that k+l<j<\V \, check whether N '(sjVjfek.
g '.
If this test fails for any ; ,
then halt; kG= k —1.
(6)
Increment k by one, go to step 2.
The correctness of the above algorithm follows from the results in Even [3].
analyze the time and space requirement o f the algorithm.
We now
W e store the graphs <7. (2<y<|V|)
along with the current flow values in the corresponding networks. In each iteration the computa­
tionally intensive steps are clearly 2 and 5. In the kih iteration, we solve it—1 maximum flow
problems in step 2 and using the flow values computed in the
k —l ih
iteration for the networks
a.
corresponding to G
we check whether N
3{ s , V j ) > k
in step 5 by finding at most one f.a.p. in each
o f the corresponding networks. Using Dinic’s algorithm [2] step 2 can be done in 0(k\E\\vf>) time
and step 5 in 0(|V||i?D time since an f.a.p. can be found in linear time. Thus the running time of
the algorithm is 0 (io| E | | v f+ i0 |V||£D = O l f c + l ^ s l E l l W 4) = 0(max(it<;j F f 4) i 0 l£||V|'4) which
is the same as Galil’s algorithm. However, the space requirement is only 0(|V|jZ?|) because the
flow values for at most |F| maximum flow problems have to be stored and each requires 0(|2?|)
space.
A c k n o w le d g m e n t . The authors wish to thank V. Ramachandran for introducing us to the
problem in a course on graph algorithms and for her many helpful suggestions during the
preparation o f this report.
REFERENCES
1.
Becker, M. et. al. A probabilistic algorithm for vertex connnectivity o f graphs.
I n fo r m . P r o c . L e t t . (1982) vol. 15, pp. 135-136.
2.
Dinic, E. A. Algorithm for solution of a problem o f maximum flow in a network with
power estimation. S o v ie t M a th . D o k l. (1970) vol. 11, pp. 1277-1280.
3.
Even, S. A n algorithm for determining whether the connectivity o f a graph is at least k.
S IA M J o u r n a l o f C o m p u tin g (1975) vol. 4, pp. 393-396.
4.
5.
. G r a p h A lg o r ith m s . Computer Science Press, Rockville, MD, 1979.
Even, S. and R. E. Tarjan. Network flow and testing graph connectivity. S IA M J o u r ­
n a l o f C o m p u t in g (1975) vol. 4, pp. 507-518.
Ford, L. R. and D. R. Fulkerson. F lo w s in N e tw o r k s . Princeton Press, Princeton,
NJ, 1962.
Galil, Z. Finding the vertex connectivity o f graphs. S IA M J o u r n a l o f C o m p u t in g
(February 1980) vol. 9, pp. 197-199.
Hopcroft, J. E. and R. E. Tarjan. Dividing a graph into triconnected components.
S IA M J o u r n a l o f C o m p u t in g (1973) pp. 135-158.
Kanevsky, A.
and V .
Ramachandran.
"Improved
algorithms
for graph fo u r-
connectivity", W orking Paper 87-14, Coordinated Science Labaratory, University
o f Illinois at Urbana-Champaign, Urbana, EL, 1987.
Linial, N., L. Lovasz and A. Wigderson. A physical interpretation o f graph connec­
tivity, and its algorithmic applications. P r o c . 2 8 th IE E E A n n . S y m p . o n
F o u n d a tio n s o f C o m p . S ci. (1985) pp. 464-467.
Miller, G. L. and V. Ramachandran. A new graph triconnectivity algorithm and its
parallelization. P r o c . o f th e 1 9 th A n n u a l A C M S y m p o s iu m o n T h e o r y o f
C o m p u t in g (May 1987).
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