Complexity of finding embeddings in a k-tree
SIAM Journal on Algebraic and Discrete Methods
Implicit representation of graphs
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Universal graphs and induced-universal graphs
Journal of Graph Theory
Finding approximate separators and computing tree width quickly
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
SIAM Journal on Computing
Embedding planar graphs on the grid
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
Proceedings of the thirteenth annual ACM symposium on Parallel algorithms and architectures
Low Redundancy in Static Dictionaries with Constant Query Time
SIAM Journal on Computing
Small Induced-Universal Graphs and Compact Implicit Graph Representations
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Excluding any graph as a minor allows a low tree-width 2-coloring
Journal of Combinatorial Theory Series B
Edge partition of planar sraphs into two outerplanar graphs
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Labeling Schemes for Small Distances in Trees
SIAM Journal on Discrete Mathematics
Compact Labeling Scheme for Ancestor Queries
SIAM Journal on Computing
Proper minor-closed families are small
Journal of Combinatorial Theory Series B
Short labels by traversal and jumping
SIROCCO'06 Proceedings of the 13th international conference on Structural Information and Communication Complexity
Localized and compact data-structure for comparability graphs
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Labeling schemes for vertex connectivity
ACM Transactions on Algorithms (TALG)
Compact navigation and distance oracles for graphs with small treewidth
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
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Implicit representation of graphs is a coding of the structure of graphs using distinct labels so that adjacency between any two vertices can be decided by inspecting their labels alone. All previous implicit representations of planar graphs were based on the classical three forests decomposition technique (a.k.a. Schnyder's trees), yielding asymptotically to a 3 log n-bit label representation where n is the number of vertices of the graph. We propose a new implicit representation of planar graphs using asymptotically 2 log n-bit labels. As a byproduct we have an explicit construction of a graph with n2+o(1) vertices containing all n-vertex planar graphs as induced subgraph, the best previous size of such induced-universal graph was O(n3). More generally, for graphs excluding a fixed minor, we construct a 2 log n + O(log log n) implicit representation. For treewidth-k graphs we give a log n + O(k log log(n/k)) implicit representation, improving the O(k log n) representation of Kannan, Naor and Rudich [18] (STOC '88). Our representations for planar and treewidth-k graphs are easy to implement, all the labels can be constructed in O(n log n) time, and support constant time adjacency testing.