On Reachability in Graphs with Bounded Independence Number
نویسندگان
چکیده
We study the reachability problem for finite directed graphs whose independence number is bounded by some constant k. This problem is a generalisation of the reachability problem for tournaments. We show that the problem is first-order definable for all k. In contrast, the reachability problems for many other types of finite graphs, including dags and trees, are not first-order definable. Also in contrast, first-order definability does not carry over to the infinite version of the problem. We prove that the number of strongly connected components in a graph with bounded independence number can be computed using TC-circuits, but cannot be computed using AC-circuits. We also study the succinct version of the problem and show that it is Π 2 -complete for all k.
منابع مشابه
A Logspace Approximation Scheme for the Shortest Path Problem for Graphs with Bounded Independence Number
How difficult is it to find a path between two vertices in finite directed graphs whose independence number is bounded by some constant k? The independence number of a graph is the largest number of vertices that can be picked such that there is no edge between any two of them. The complexity of this problem depends on the exact question we ask: Do we only wish to tell whether a path exists? Do...
متن کاملThe Descriptive Complexity of the Reachability Problem As a Function of Different Graph Parameters
The reachability problem for graphs cannot be described, in the sense of descriptive complexity theory, using a single first-order formula. This is true both for directed and undirected graphs, both in the finite and infinite. However, if we restrict ourselves to graphs in which a certain graph parameter is fixed to a certain value, first-order formulas often suffice. A trivial example are grap...
متن کاملDistributed Large Independent Sets in One Round on Bounded-Independence Graphs
We present a randomized one-round, single-bit messages, distributed algorithm for the maximum independent set problem in polynomially bounded-independence graphs with poly-logarithmic approximation factor. Bounded-independence graphs capture various models of wireless networks such as the unit disc graphs model or the quasi unit disc graphs model. For instance, on unit disc graphs, our achieved...
متن کاملA comparison of two approaches for polynomial time algorithms computing basic graph parameters
In this paper we compare and illustrate the algorithmic use of graphs of bounded treewidth and graphs of bounded clique-width. For this purpose we give polynomial time algorithms for computing the four basic graph parameters independence number, clique number, chromatic number, and clique covering number on a given tree structure of graphs of bounded tree-width and graphs of bounded clique-widt...
متن کاملBounded Depth Arithmetic Circuits: Counting and Closure
Constant-depth arithmetic circuits have been defined and studied in [AAD97, ABL98]; these circuits yield the function classes #AC and GapAC. These function classes in turn provide new characterizations of the computational power of threshold circuits, and provide a link between the circuit classes AC (where many lower bounds are known) and TC (where essentially no lower bounds are known). In th...
متن کامل