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Implicit Representation of Graphs

Implicit Representation of Graphs. Paper by Sampath Kannan, Moni Naor, Steven Rudich. Introduction. Def.: A vertex induced subgraph or simply an induced subgraph G` of G is a vertex set together with the edge set Def.: K-labeling scheme F family of finite graphs

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Implicit Representation of Graphs

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  1. Implicit Representation of Graphs Paper by Sampath Kannan, Moni Naor, Steven Rudich

  2. Introduction • Def.: A vertex induced subgraph or simply an induced subgraph G` of G is a vertex set together with the edge set • Def.: K-labeling scheme • F family of finite graphs • A graph G in F of n vertices • Each vertex label no more than k logn bits • Deciding adjacency in polynomial time of label length

  3. Introduction • Def.: Vertex Induced Universal Graph • S a finite set of graphs • G is vertex induced universal for S if every graph in S is a vertex induced subgraph of G • Def.: F a family of graphs has universal graphs of size g(n) • For every n there is a universal graph of size less than or equal g(n) for all graphs in F with n or fewer vertices

  4. Labelable Families • Proposition: A family of graphs that contains more than n-vertex graphs cannot be labeled • O(nlogn) bits to represent n-vertex graph • At most can be represented

  5. Labelable Families (cont.) • Finite trees( and finite forests) • 2-labeling scheme • Arbitrarily prelabel the vertices from 1 to n • For each vertex label concatenate It’s own prelabel and its parent’s prelabel (2logn bits) • Adjacency: check the first half of one label with the second half of the other

  6. Labelable Families (cont.) • Transitive Closure of Trees: • Ancestorhood relation • 2-labeling scheme • Let T a tree and T` the transitive closure of T • Traverse T in post-order • For each vertex assign the interval between its smallest numbered descendant and its largest one • Ancestorhood: u ancestor of v iff u’s interval contains v’s interval T T`

  7. Labelable Families (cont.) • Sparse Graphs or Graphs with Bounded Arboricity: • the minimum number of forests into which its edges can be partitioned • the minimum number of spanning trees needed to cover all the edges of the graph • the subgraphs of any graph cannot have arboricity larger than the graph itself, or equivalently the arboricity of a graph must be at least the maximum arboricity of any of its subgraph • H induced subgraph of G

  8. Labelable Families (cont.) • Graphs with Bounded Arboricity • K+1 labeling scheme for graphs of arboricity k • Prelabel the vertices arbitrarily • Decompose the graph into K forests • Concatenate to each vertex label the label of its parent in each of the k forests • Adjacency: check if one vertex is the parent of the other in any of the forests

  9. Labelable Families (cont.) • Graphs of bounded degree d have arboricity bounded by • Planar graphs have arboricity of 3  4-labeling scheme

  10. Labelable Families • Intersection Graph: A graph where vertices represent sets • Edge exist if two sets intersect • Interval graph • Path graph: • Each vertex represent a path in a tree • Two vertices adjacent iff the paths representing them intersect

  11. 1-3 1-2 3 1 2 Labelable Families (cont.) • Path Graphs: • Label the transitive closure T` of the tree T containing all paths • Label of each vertex in the path graph consists of : • Label of the beginning vertex in T` • Label of the apex vertex in T` • Label of the end vertex in T` • Adjacency: test if the apex of one path vertex is sandwiched between the apex and an end of the other path

  12. c-Decomposable Graphs • A graph G is c-decomposable if for all subgraphs H with no more than c vertices there exist c vertices s.t. their removal causes H to be disconnected with no component of size 2|H|/3 • Construct T a tree decomposition of G • Chose a c-separator as the root of T • Each component will be a subtree of root • Each vertex v of G occurs in a vertex t(v) in T • Each vertex of T is assigned at most c vertices • The depth of T is at most

  13. c-Decomposable Graphs (cont.) • The label of each vertex v in G consists of: • the path in T from the root to t(v) • The rank of v in t(v) • For each vertex s of the tree along the path from root to t(v), a c-bit vector giving adj. info. between v and those of s • Adjacency: To test adj. between u and v in G • Check if t(u) is ancestor of t(v) or vice versa • Determine the depth of i of the ancestor say u • Check the i’th c-vector of v at the position of u’s rank

  14. Labeling Schemes and Universal Graphs • Thm: If a family F has k-labeling scheme, then it has universal graphs of size nk constructible in polynomial time. • Proof: • Form Graph U with vertices labeled 1 to nk • place an edge between two vertices if their adjacency test produces true • E.g: universal graph for tree • n2 vertices of U • Label vertices of U as (i,j)where 1<= i,j <= n • Add edge between two vertices u(i,j) and v=(i`,j`) if i=j`or j= i`

  15. References •   S. Kannan, M. Naor, S. Rudich, Implicit representation of graphs, Proceedings of the twentieth annual ACM symposium on Theory of Computing,  Pages: 334 - 343  (1988). • Wikipedia:http://www.wikipedia.org for figures

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