52 research outputs found
Randomized Communication and Implicit Representations for Matrices and Graphs of Small Sign-Rank
We prove a characterization of the structural conditions on matrices of
sign-rank 3 and unit disk graphs (UDGs) which permit constant-cost public-coin
randomized communication protocols. Therefore, under these conditions, these
graphs also admit implicit representations.
The sign-rank of a matrix is the smallest rank
of a matrix such that for all ; equivalently, it is the smallest dimension in which can be
represented as a point-halfspace incidence matrix with halfspaces through the
origin, and it is essentially equivalent to the unbounded-error communication
complexity. Matrices of sign-rank 3 can achieve the maximum possible
bounded-error randomized communication complexity , and
meanwhile the existence of implicit representations for graphs of bounded
sign-rank (including UDGs, which have sign-rank 4) has been open since at least
2003. We prove that matrices of sign-rank 3, and UDGs, have constant randomized
communication complexity if and only if they do not encode arbitrarily large
instances of the Greater-Than communication problem, or, equivalently, if they
do not contain arbitrarily large half-graphs as semi-induced subgraphs. This
also establishes the existence of implicit representations for these graphs
under the same conditions.Comment: 28 page
Succinct Permutation Graphs
We present a succinct, i.e., asymptotically space-optimal, data structure for permutation graphs that supports distance, adjacency, neighborhood and shortest-path queries in optimal time; a variant of our data structure also supports degree queries in time independent of the neighborhood's size at the expense of an -factor overhead in all running times. We show how to generalize our data structure to the class of circular permutation graphs with asymptotically no extra space, while supporting the same queries in optimal time. Furthermore, we develop a similar compact data structure for the special case of bipartite permutation graphs and conjecture that it is succinct for this class. We demonstrate how to execute algorithms directly over our succinct representations for several combinatorial problems on permutation graphs: Clique, Coloring, Independent Set, Hamiltonian Cycle, All-Pair Shortest Paths, and others. Moreover, we initiate the study of semi-local graph representations; a concept that "interpolates" between local labeling schemes and standard "centralized" data structures. We show how to turn some of our data structures into semi-local representations by storing only bits of additional global information, beating the lower bound on distance labeling schemes for permutation graphs
Sharp Thresholds in Random Simple Temporal Graphs
A graph whose edges only appear at certain points in time is called a temporal graph (among other names). Such a graph is temporally connected if each ordered pair of vertices is connected by a path which traverses edges in chronological order (i.e., a temporal path). In this paper, we consider a simple model of random temporal graph, obtained from an Erd\H{o}s-R\'enyi random graph by considering a random permutation of the edges and interpreting the ranks in as presence times. Temporal reachability in this model exhibits a surprisingly regular sequence of thresholds. In particular, we show that at any fixed pair of vertices can a.a.s. reach each other; at at least one vertex (and in fact, any fixed vertex) can a.a.s. reach all others; and at all the vertices can a.a.s. reach each other, i.e., the graph is temporally connected. Furthermore, the graph admits a temporal spanner of size as soon as it becomes temporally connected, which is nearly optimal as is a lower bound. This result is significant because temporal graphs do not admit spanners of size in general (Kempe et al, STOC 2000). In fact, they do not even admit spanners of size (Axiotis et al, ICALP 2016). Thus, our result implies that the obstructions found in these works, and more generally, all non-negligible obstructions, must be statistically insignificant: nearly optimal spanners always exist in random temporal graphs. All the above thresholds are sharp. Carrying the study of temporal spanners further, we show that pivotal spanners -- i.e., spanners of size made of two spanning trees glued at a single vertex (one descending in time, the other ascending subsequently) -- exist a.a.s. at , this threshold being also sharp. Finally, we show that optimal spanners (of size ) also exist a.a.s. at
On Forbidden Induced Subgraphs for Unit Disk Graphs
A unit disk graph is the intersection graph of disks of equal radii in the plane. The class of unit disk graphs is hereditary, and therefore admits a characterization in terms of minimal forbidden induced subgraphs. In spite of quite active study of unit disk graphs very little is known about minimal forbidden induced subgraphs for this class. We found only finitely many minimal non-unit disk graphs in the literature. In this paper we study in a systematic way forbidden induced subgraphs for the class of unit disk graphs. We develop several structural and geometrical tools, and use them to reveal infinitely many new minimal non-unit disk graphs. Further we use these results to investigate structure of co-bipartite unit disk graphs. In particular, we give structural characterization of those co-bipartite unit disk graphs whose edges between parts form a C4-free bipartite graph, and show that bipartite complements of these graphs are also unit disk graphs. Our results lead us to propose a conjecture that the class of co-bipartite unit disk graphs is closed under bipartite complementation
Graphs with minimum fractional domatic number
The domatic number of a graph is the maximum number of vertex disjoint
dominating sets that partition the vertex set of the graph. In this paper we
consider the fractional variant of this notion. Graphs with fractional domatic
number 1 are exactly the graphs that contain an isolated vertex. Furthermore,
it is known that all other graphs have fractional domatic number at least 2. In
this note we characterize graphs with fractional domatic number 2. More
specifically, we show that a graph without isolated vertices has fractional
domatic number 2 if and only if it has a vertex of degree 1 or a connected
component isomorphic to a 4-cycle. We conjecture that if the fractional domatic
number is more than 2, then it is at least 7/3
Faster Exploration of Some Temporal Graphs
A temporal graph G = (G_1, G_2, ..., G_T) is a graph represented by a sequence of T graphs over a common set of vertices, such that at the i-th time step only the edge set E_i is active. The temporal graph exploration problem asks for a shortest temporal walk on some temporal graph visiting every vertex. We show that temporal graphs with n vertices can be explored in O(k n^{1.5} log n) days if the underlying graph has treewidth k and in O(n^{1.75} log n) days if the underlying graph is planar. Furthermore, we show that any temporal graph whose underlying graph is a cycle with k chords can be explored in at most 6kn days. Finally, we demonstrate that there are temporal realisations of sub cubic planar graphs that cannot be explored faster than in ?(n log n) days. All these improve best known results in the literature
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