In the context of efficient representations of graphs, J. H. Muller defined a local structure or adjacency labeling
scheme for a graph G in a given family F of graphs to be an assignment of an O(log n)-bit identifier to each vertex
of G, together with an algorithm (that may depend on F but is independent of the individual graph G) that takes as input two vertex identifiers and determines whether or not they are the endpoints of an edge in G. That is, this type of implicit representation is analogous to an adjacency matrix: it is straightforward to check whether two vertices are
adjacent but finding the neighbors of any vertex requires a search through all possible vertices.[5]
Graph families with adjacency labeling schemes include:
• Sparse graphs. If every vertex in G has at most d neighbors, one may number the vertices of G from 1 to n and let the identifier for a vertex be the (d + 1)-tuple of its own number and the numbers of its neighbors. Two
Implicit graph 39 vertices are adjacent when the first numbers in their identifiers appear later in the other vertex's identifier. More generally, the same approach can be used to provide an implicit representation for graphs with bounded arboricity
or bounded degeneracy, including the planar graphs and the graphs in any minor-closed graph family.[6][7]
• Intersection graphs. An interval graph is the intersection graph of a set of line segments in the real line. It may be given an adjacency labeling scheme in which the points that are endpoints of line segments are numbered from 1 to 2n and each vertex of the graph is represented by the numbers of the two endpoints of its corresponding interval. With this representation, one may check whether two vertices are adjacent by comparing the numbers that represent them and verifying that these numbers define overlapping intervals. The same approach works for other geometric intersection graphs including the graphs of bounded boxicity and the circle graphs, and
subfamilies of these families such as the distance-hereditary graphs and cographs.[6][8] However, a geometric
intersection graph representation does not always imply the existence of an adjacency labeling scheme, because it may require more than a logarithmic number of bits to specify each geometric object; for instance, representing a
graph as a unit disk graph may require exponentially many bits for the coordinates of the disk centers.[9]
• Low-dimensional comparability graphs. The comparability graph for a partially ordered set has a vertex for each set element and an edge between two set elements that are related by the partial order. The order dimension of a partial order is the minimum number of linear orders whose intersection is the given partial order. If a partial order has bounded order dimension, then an adjacency labeling scheme for the vertices in its comparability graph may be defined by labeling each vertex with its position in each of the defining linear orders, and determining that two vertices are adjacent if each corresponding pair of numbers in their labels has the same order relation as each other pair. In particular, this allows for an adjacency labeling scheme for the chordal comparability graphs, which
come from partial orders of dimension at most four.[10][11]
Not all graph families have local structures. For some families, a simple counting argument proves that adjacency labeling schemes do not exist: only O(n log n) bits may be used to represent an entire graph, so a representation of
this type can only exist when the number of n-vertex graphs in the given family F is at most 2O(n log n). Graph
families that have larger numbers of graphs than this, such as the bipartite graphs or the triangle-free graphs, do not
have adjacency labeling schemes.[6] [8] However, even families of graphs in which the number of graphs in the
family is small might not have an adjacency labeling scheme; for instance, the family of graphs with fewer edges
than vertices has 2O(n log n) n-vertex graphs but does not have an adjacency labeling scheme, because one could
transform any given graph into a larger graph in this family by adding a new isolated vertex for each edge, without
changing its labelability.[5][8] Kannan et al. asked whether having a forbidden subgraph characterization and having
at most 2O(n log n) n-vertex graphs are together enough to guarantee the existence of an adjacency labeling scheme;
this question, which Spinrad restated as a conjecture, remains open.[6][8]
If a graph family F has an adjacency labeling scheme, then the n-vertex graphs in F may be represented as induced subgraphs of a common universal graph of polynomial size, the graph consisting of all possible vertex identifiers. Conversely, if a universal graph of this type can be constructed, then the identities of its vertices may be used as
labels in an adjacency labeling scheme.[6] For this application of implicit graph representations, it is important that
the labels use as few bits as possible, because the number of bits in the labels translates directly into the number of
vertices in the universal graph. Alstrup and Rauhe showed that any tree has an adjacency labeling scheme with log2 n
+ O(log* n) bits per label, from which it follows that any graph with arboricity k has a scheme with k log2 n + O(log*
n) bits per label and a universal graph with nk2O(log*n) vertices. In particular, planar graphs have arboricity at most
Implicit graph 40
Evasiveness
The Aanderaa–Karp–Rosenberg conjecture concerns implicit graphs given as a set of labeled vertices with a black-box rule for determining whether any two vertices are adjacent; this differs from an adjacency labeling scheme in that the rule may be specific to a particular graph rather than being a generic rule that applies to all graphs in a family. This difference allows every graph to have an implicit representation: for instance, the rule could be to look up the pair of vertices in a separate adjacency matrix. However, an algorithm that is given as input an implicit graph of this type must operate on it only through the implicit adjacency test, without reference to the implementation of that test.
A graph property is the question of whether a graph belongs to a given family of graphs; the answer must remain invariant under any relabeling of the vertices. In this context, the question to be determined is how many pairs of vertices must be tested for adjacency, in the worst case, before the property of interest can be determined to be true or false for a given implicit graph. Rivest and Vuillemin proved that any deterministic algorithm for any nontrivial
graph property must test a quadratic number of pairs of vertices;[13] the full Aanderaa–Karp–Rosenberg conjecture is
that any deterministic algorithm for a monotonic graph property (one that remains true if more edges are added to a graph with the property) must in some cases test every possible pair of vertices. Several cases of the conjecture have
been proven to be true—for instance, it is known to be true for graphs with a prime number of vertices[14] —but the
full conjecture remains open. Variants of the problem for randomized algorithms and quantum algorithms have also been studied.
Bender and Ron have shown that, in the same model used for the evasiveness conjecture, it is possible in only constant time to distinguish directed acyclic graphs from graphs that are very far from being acyclic. In contrast,
such a fast time is not possible in neighborhood-based implicit graph models,[15]
References
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[2] Papadimitriou, Christos (1994), "On the complexity of the parity argument and other inefficient proofs of existence" (http://www.cs. berkeley.edu/~christos/papers/On the Complexity.pdf), Journal of Computer and System Sciences 48 (3): 498–532,
doi:10.1016/S0022-0000(05)80063-7,
[3] Immerman, Neil (1999), "Exercise 3.7 (Everything is a Graph)" (http://books.google.com/books?id=kWSZ0OWnupkC&pg=PA48),
Descriptive Complexity, Graduate Texts in Computer Science, Springer-Verlag, p. 48, ISBN 9780387986005, .
[4] Childs, Andrew M.; Cleve, Richard; Deotto, Enrico; Farhi, Edward; Gutmann, Sam; Spielman, Daniel A. (2003), "Exponential algorithmic speedup by a quantum walk", Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, New York: ACM, pp. 59–68, doi:10.1145/780542.780552, MR2121062.
[5] Muller, John Harold (1988), Local structure in graph classes, Ph.D. thesis, Georgia Institute of Technology.
[6] Kannan, Sampath; Naor, Moni; Rudich, Steven (1992), "Implicit representation of graphs", SIAM Journal on Discrete Mathematics 5 (4): 596–603, doi:10.1137/0405049, MR1186827.
[7] Chrobak, Marek; Eppstein, David (1991), "Planar orientations with low out-degree and compaction of adjacency matrices" (http://www.ics. uci.edu/~eppstein/pubs/ChrEpp-TCS-91.pdf), Theoretical Computer Science 86 (2): 243–266, doi:10.1016/0304-3975(91)90020-3, . [8] Spinrad, Jeremy P. (2003), "2. Implicit graph representation" (http://books.google.com/books?id=RrtXSKMAmWgC&pg=PA17),
Efficient Graph Representations, pp. 17–30, ISBN 0-8218-2815-0, .
[9] Kang, Ross J.; Müller, Tobias (2011), Sphere and dot product representations of graphs (http://homepages.cwi.nl/~mueller/Papers/ SphericityDotproduct.pdf), .
[10] Ma, Tze Heng; Spinrad, Jeremy P. (1991), "Cycle-free partial orders and chordal comparability graphs", Order 8 (1): 49–61, doi:10.1007/BF00385814, MR1129614.
[11] Curtis, Andrew R.; Izurieta, Clemente; Joeris, Benson; Lundberg, Scott; McConnell, Ross M. (2010), "An implicit representation of chordal comparability graphs in linear time", Discrete Applied Mathematics 158 (8): 869–875, doi:10.1016/j.dam.2010.01.005, MR2602811. [12] Alstrup, Stephen; Rauhe, Theis (2002), "Small induced-universal graphs and compact implicit graph representations" (http://www.it-c.dk/
research/algorithms/Kurser/AD/2002E/Uge7/parent.pdf), Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer
Science: 53–62, doi:10.1109/SFCS.2002.1181882, .
[13] Rivest, Ronald L.; Vuillemin, Jean (1975), "A generalization and proof of the Aanderaa-Rosenberg conjecture", Proc. 7th ACM Symposium
Implicit graph 41
[14] Kahn, Jeff; Saks, Michael; Sturtevant, Dean (1983), "A topological approach to evasiveness", Symposium on Foundations of Computer
Science, Los Alamitos, CA, USA: IEEE Computer Society, pp. 31–33, doi:10.1109/SFCS.1983.4.
[15] Bender, Michael A.; Ron, Dana (2000), "Testing acyclicity of directed graphs in sublinear time", Automata, languages and programming
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