Andrzej Proskurowski

Complexity of finding embeddings in a k -tree

A k-tree is a graph that can be reduced to the k-complete graph by a sequence of removals of a degree k vertex with completely connected neighbors. We address the problem of determining whether a graph is a partial graph of a k-tree. This problem is motivated by the existence of polynomial time algorithms for many combinatorial problems on graphs when the graph is constrained to be a partial k-tree for fixed k. These algorithms have practical applications in areas such as reliability, concurrent broadcasting and evaluation of queries in a relational database system. We determine the complexity status of two problems related to finding the smallest number k such that a given graph is a partial k-tree. First, the corresponding decision problem is NP-complete. Second, for a fixed (predetermined) value of k, we present an algorithm with polynomially bounded (but exponential in k) worst case time complexity. Previously, this problem had only been solved for

Linear time algorithms for NP-hard problems restricted to partial k -trees

k = 1,2,3

An algebraic theory of graph reduction

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Characterization and recognition of partial 3-trees

Abstract We present and illustrate by a sequence of examples an algorithm paradigm for solving NP- hard problems on graphs restricted to partial graphs of k -trees and given with an embedding in a k -tree. Such algorithms, linear in the size of the graph but exponential or superexponential in k , exist for most NP-hard problems that have linear time algorithms for trees. The examples used are optimization problems involving independent sets, dominating sets, graph coloring, Hamiltonian circuits, network reliability and minimum vertex deletion forbidden subgraphs. The results generalize previous results for series-parallel graphs, bandwidth-constrained graphs, and non- serial dynamic programming.

Minimum broadcast graphs

We show how membership in classes of graphs definable in monadic second order logic and of bounded treewidth can be decided by finite sets of terminating reduction rules. The method is constructive in the sense that we describe an algorithm which will produce, from a formula in monadic second order logic and an integer k such that the class defined by the formula is of treewidth ≤ k, a set of rewrite rules that reduces any member of the class to one of finitely many graphs, in a number of steps bounded by the size of the graph. This reduction system corresponds to an algorithm that runs in time linear in the size of the graph.

Forbidden minors characterization of partial 3-trees

Our interest in the class of k-trees and their partial graphs and subgraphs is motivated by some practical questions about the reliability of communication networks in the presence of constrained line- and site-failures, and about the complexity of queries in a data base system. We have found a set of confluent graph reductions such that any graph can be reduced to the empty graph if and only if it is a subgraph of a 3-tree. This set of reductions yields a polynomial time algorithm for deciding if a given graph is a partial 3-tree and for finding one of its embeddings in a 3-tree when such an embedding exists. Our result generalizes a previously known recognition algorithm for partial 2-trees (series-parallel graphs).

Practical Algorithms on Partial k-Trees with an Application to Domination-like Problems

Let a graph G = (V, E) represent a communication network. The set V of vertices corresponds to the members of the network, and the set E of edges corresponds to the communication lines connecting pairs of members. Suppose that a member originates a message which is to be communicated to all other members of the network. This is to be accomplished as quickly as possible by a series of calls placed over lines of the network. We adopt the constraints that (i) each call requires one unit of time, (ii) a member can participate in only one call per unit of time, and (iii) a member can only call an adjacent member. We refer to this one-to-all communication process as broadcasting. Consider the following problem: given a connected graph G and a meSSage originator, vertex u, what is the minimum number of time units required to complete broadcasting from vertex u? We define the broadcast time of u vertex u, b(u), to equal this minimum time. It is easy to see that for any vertex u in a connected graph G with n vertices, b(u) a [ log, nl , since during each time unit the number of informed vertices can at most double. We define the broadcast time of u graph G, 6(G), to equal the maximum broadcast time of any vertex u in G, i.e., b(G) = max {6(u) 1 u E V(G)}. For the complete graph K, with n > 2 vertices, b(K,) = [ log, nl , yet K, is not minimal with respect to this property. Th’at is, we can remove several edges from & and still have a granh G such that b(G) = [ log, nl . We define a minimal broadcast graph to be a graph G with n vertices such that b(G) = [log, nl but for every proper spanning subgraph G’c G, b( G’) > [log* 4 l We define the broadcast function B(n) to equal the minimum number of edges in any minimal broadcast graph on n vertices. A minimum broadcast graph is a minimal broadcast graph on n vertices having B(n) edges. Frolm the point of view of applications, minimum broadcast graphs represent the cheapest possible cammunication networks (in terms of number of lines) in which broadcasting can be accomplished from any vertex as fast as theoretically possible.

On the Generation of Binary Trees

Abstract A k -tree is formed from a k -complete graph by recursively adding a vertex adjacent to all vertices in an existing k -complete subgraph. The many applications of partial k -trees (subgraphs of k -trees) have motivated their study from both the algorithmic and theoretical points of view. In this paper we characterize the class of partial 3-trees by its set of four minimal forbidden minors ( H is a minor of G if H can be obtained from G by a finite sequence of edge-extraction and edge-contradiction operations.)

Separating subgraphs in k-trees: Cables and caterpillars

Many NP-hard problems on graphs have polynomial, in fact usually linear, dynamic programming algorithms when restricted to partial k-trees (graphs of treewidth bounded by k), for fixed values of k. We investigate the practicality of such algorithms, both in terms of their complexity and their derivation, and account for the dependency on the treewidth k. We define a general procedure to derive the details of table updates in the dynamic programming solution algorithms. This procedure is based on a binary parse tree of the input graph. We give a formal description of vertex subset optimization problems in a class that includes several variants of domination, independence, efficiency and packing. We give algorithms for any problem in this class, which take a graph G, integer k and a width k tree-decomposition of G as input, and solve the problem on G in O(n24k) steps.

Covering Regular Graphs

A binary tree may be uniquely represented by a code reflecting traversal of the corresponding extended binary tree m a given monotomc order A general algorithm for constructing codes of all binary trees with n vertices is presented Different orders of traversal yield different ordermgs of the generated trees The algorithm IS illustrated with an example of the sequence of binary trees obtained from ballot sequences