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We prove that every n-vertex graph G with pathwidth pw(G) has a three-dimensional straight-line grid drawing with O(pw(G) n) volume. Thus for

In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A k-stack (resp...

As trees are used in a wide variety of application areas, the comparison of trees arises in many guises. Here we consider two generalizations of classical tree pattern matching, which consists of determining if one tree is isomorphic to a subgraph of another. For the embedding problems of subgraph isomorphism and topological embedding, we present algorithms for determining the largest tree embeddable in two trees T and T 0 (or a largest subtree) and for constructing the smallest tree in which each of T and T 0 can be embedded (or a smallest supertree). Both subtrees and supertrees can be used in a variety of different applications. For example, when each of the two trees contains partial information about a data set, such as the evolution of a set of species, the subtree or supertree corresponds to a structuring of the data in a manner consistent with both original trees. The size of a subtree or supertree of two trees can also be used to measure the similarity between two arrangem...

The subgraph isomorphism problem, that of nding a copy of one graph in another, has proved to be intractable except when certain restrictions are placed on the inputs. In this paper, we introduce a new property for graphs (a generalization on bounded degree) and extend the known classes of inputs for which polynomial-time subgraph isomorphism algorithms are attainable. In particular, if the removal of any set of at most k vertices from an n-vertex graph results in O(k log n) connected components, we say that the graph is a log-bounded fragmentation graph. We present a polynomial-time algorithm for nding a subgraph of H isomorphic to a graph G when G is a log-bounded fragmentation graph and H has bounded treewidth; these results are extended to handle graphs of locally bounded treewidth (a generalization of treewidth) when G is a log-bounded fragmentation graph and has constant diameter.

The problem of determining whether a k-connected partial k-tree is isomorphic to subgraph of another partial k-tree is shown to be solvable in time O(n k+2 ). The presented time-bounds considerably improve the corresponding bounds known in the literature. They rely in part on a new characterization of width-k tree-decomposition of k-connected partial k-trees. 1 Introduction The subgraph isomorphism problem is to determine whether a graph is isomorphic to a subgraph of another graph. It is a fundamental graph problem with a variety of applications in, for instance, engineering sciences, organic chemistry, biology, and pattern matching. For instance, if G is an n-vertex cycle and H is an n-vertex planar graph with each vertex of degree 3; then determining whether G is isomorphic to a subgraph of H is equivalent to the NP-complete problem of determining whether a 3-regular planar graph has a Hamiltonian circuit [6]. Thus, the subgraph isomorphism problem is NP-complete even if G...

Many real-life problems can be modeled by graph-theoretic problems. These graph problems are usually NP-hard and hence there is no efficient algorithm for solving them, unless P= NP. One way to overcome this hardness is to solve the problems when restricted to special graphs. Trees are one kind of graph for which several NP-complete problems can be solved in polynomial time. Graphs of bounded treewidth, which generalize trees, show good algorithmic properties similar to those of trees. Using ideas developed for tree algorithms, Arnborg and Proskurowski introduced a general dynamic programming approach which solves many problems such as dominating set, vertex cover and independent set. Others used this approach to solve other NP-hard problems. Matousek and Thomas applied this approach to solve the subgraph isomorphism problem when the source graph has bounded degree and the host graph has bounded treewidth. In this thesis, we introduce a new property for graphs called log-bounded fragmentation, by which we mean after removing any set of at most k vertices the number of connected components is at most O(k log n), where n is the number of vertices of the graph. We then extend the result of Matousek and Thomas to the case in which the source graph is a log-bounded fragmentation graph and the host graph has bounded treewidth. Besides this result, we demonstrate how bounded fragmentation might be used to measure the reliability of a network.

. We present O(n 3 ) embedding algorithms (generalizing subgraph isomorphism) for classes of graphs of bounded pathwidth, where n is the number of vertices in the graph. These include the rst polynomialtime algorithm for minor containment and the rst O(n c ) algorithm (c a constant independent of k) for topological embedding of graphs from subclasses of partial k-trees. Of independent interest are structural properties of k-connected graphs of bounded pathwidth on which our algorithms are based. We also describe special cases which reduce to various generalizations of string matching, permitting more ecient solutions. 1 Introduction Many fundamental problems in a diverse set of research areas can be characterized as graph embedding problems, where data is represented as graphs and patterns can be detected by nding smaller graphs in larger ones. Classic patternmatching problems make use of the subgraph isomorphism problem, namely, the problem of determining whether ther...

A hybrid algorithm is a collection of heuristics, paired with a polynomial time selector S that runs on the input to decide which heuristic should be executed to solve the problem. Hybrid algorithms are interesting in scenarios where the selector must decide between heuristics that are “good ” with respect to different complexity measures. In this paper, we focus on hybrid algorithms with a “hardness-defying ” property: for a problem Π, there is a set of complexity measures {mi} whereby Π is known or conjectured to be hard (or unsolvable) for each mi, but for each heuristic hi of the hybrid algorithm, one can give a complexity guarantee for hi on the instances of Π that S selects for hi that is strictly better than mi. For example, we show that for NP-hard problems such as Max-Ek-Lin-p, Longest Path and Minimum Bandwidth, a given instance can either be solved exactly in “sub-exponential ” (2 o(n)) time, or be approximated in polynomial time with an approximation ratio exceeding that of the known or conjectured inapproximability of the problem, assuming P ̸ = NP. We also prove some inherent limitations to the design of hybrid algorithms that arise under the assumption that NP

Documents can be represented as structures with a hierarchial arrangement of text and non-text nodes, where nodes are labeled by category names such as paragraph and section. Representing documents this way is a natural consequence of using the Standard Generalized Markup Language(SGML) to encode text documents which has many applications in di#erent areas.