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Sub Graph Listing

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 : a cycle of length three.(C3)

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Sub Graph Listing

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  1. Sub Graph Listing محمد مهدی طالبی دانشگاه صنعتی امیرکبیر

  2. 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.

  3. 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.

  4. 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 subgraphH 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.

  5. 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

  6. 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 .

  7. 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.

  8. 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.

  9. Listing quadrangles • _ • _ • _

  10. Listing maximal cliques • _ • _

  11. =3 y = 2 = 3 N(y) =3

  12. 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.

  13. Thanks

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