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 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 n k constructible in polynomial time. Proof:  Form Graph U with vertices labeled 1 to n k  place an edge between two vertices if their adjacency test produces true E.g: universal graph for tree n 2 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: (1988). Wikipedia: for figures