Sub Graph Listingمحمد مهدی طالبی

دانشگاه صنعتی امیرکبیر

spanning subgraph of G : is a subgraph of G which includes all the vertices of G.

a spanning tree of G : is a spanning subgraph of G which is a tree.

Triangle : a cycle of length three.(C3)

Maximal Clique: maximal complete subgraph in G.

a(G) : the minimum number of edge disjoint spanning forests into which G can be decomposed.

Lemma 8.1. Let G be a graph, then (m:edges, n:vertices)

Proof.Nash-Williams[Nas61] showed that

Suppose that the maximum in the right-hand side of (8.2) is achieved by a subgraph H having p vertices and q edges. Let k be the number of edges in a clique with p vertices, that is, k = p ( p - 1)/2.

It should be noted that a(G) = O(1) for a large class of graphs including:•planar graphs • graphs of bounded genus• graphs of bounded maximum degree

Lemma 8.2.

Proof. : the edge-disjoint spanning forests of G such ThatAssociate each edge of with a vertex of G as follows:choose an arbitrary vertex u of each tree T in forest as the root of T;regard T as a rooted tree with root u in which all the edges are directed from theroot to the descendants; and associate each edge e of tree T with the headvertex h(e) of e. Thus, every vertex of except the roots, is associated withexactly one edge of .

Listing TrianglesThe triangle detection problem often arises in many combinatorial

problems such as: the minimum cycle detection problem [IR78] the approximate Hamiltonian walk problem in maximal planar

graphs [NAW83] the approximate minimum vertex cover (or maximum independent

set) problem in planar graphs in [Alb74, BE821]

[IR78] : spends space and runs in time for general graphs and in O(n) time for planar graphs.

[BE821]: improved the space complexity of the algorithm from into O(n) by avoiding the use of the adjacency matrix.

2O(n ) 3/2O(m )

2O(n )

Clearly the degrees of vertices can be computed in O(m) time. Since the degree of any vertex is at most n - 1, one can sort the

vertices in O(n) time by the bucket sort. Using adjacency lists, we can delete a vertex v from G in O(d(v)

time, and scan all the vertices adjacent to a vertex v in O(d(v)) time.

The time required by the ith iteration of the outmost for statement Statements 1, 3 and 4 spend O(d(vi)) time. Statement 2 requires at most time.

If G is planar, the algorithm runs in O(a(G)m)<O (n) time since a(G)< 3 by Lemma 8.l.

Listing quadrangles_

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Listing maximal cliques_

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=3 y = 2 = 3 N(y) =3yC0C

Bipartite Subgraph Listing Algorithms Given any undirected graph G, a d-bounded orientation of G is

simply an orientation in which each vertex has out-degree at most d.

An acyclic orientation is one in which there is no directed cycle. An advantage of acyclic orientations is that they are easy to

construct.

Thanks