Sub Graph Listing محمد 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 Triangle : a cycle of length three.(C3) Maximal Clique Maximal Clique : maximal complete subgraph in G. a(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 planar graphs graphs of bounded genus graphs of bounded genus graphs of bounded maximum degree graphs of bounded maximum degree

Lemma 8.2. Proof. : the edge-disjoint spanning forests of G such That Associate 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 the root to the descendants; and associate each edge e of tree T with the head vertex h(e) of e. Thus, every vertex of except the roots, is associated with exactly one edge of.

Listing Triangles The 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.

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 _

Listing maximal cliques _

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

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