Graph Searches and Their End Vertices.

A polynomial algorithm for the k-cluster problem on the interval graphs

Partitioning subclasses of chordal graphs with few deletions

Many problems in real-world applications require searching graphs. Understanding the performance of search algorithms has been one of the eminent tasks of heuristic search research. Despite the importance of graph search algorithms, the research of analyzing their performance is limited, and most work on search algorithm analysis has been focused on tree search algorithms. One of the major obstacles to analyzing graph search is that no single graph is an appropriate representative of graph search problems. In this paper, we propose one possible approach to analyzing graph search: Analyzing the performance of graph search algorithms on a representative graph of a cluster of problems. We specifically consider job-sequencing problems in which a set of jobs must be sequenced on a machine such that a penalty function is minimized. We analyze the performance of A* graph search algorithm on an abstract model that closely represents job sequencing problems. It is an extension to a model widely used previously for analyzing tree search. One of the main results of our analysis is the existence of a gap of computational cost between two classes of job sequencing problems, one with exponential and the other with polynomial complexity. We provide experimental results showing that real job sequencing problems indeed have a huge difference on computational costs under different conditions.

Graph searching is one of the simplest and most widely used tools in graph algorithms. Every graph search method is defined using some partic-ular selection rule, and the analysis of the corre-sponding vertex orderings can aid greatly in de-vising algorithms, writing proofs of correctness, or recognition of various graph families. We study graphs where the sets of vertex order-ings produced by two di˙erent search methods coincide. We characterise such graph families for ten pairs from the best-known set of graph searches: Breadth First Search (BFS), Depth First Search (DFS), Lexicographic Breadth First Search (LexBFS) and Lexicographic Depth First Search (LexDFS), and Maximal Neighborhood Search (MNS).

The edge-orientation problem and some of its variants on weighted graphs

Graph Problems Related to Gate Matrix Layout and PLA Folding. This paper gives a survey on graph problems occuring in linear VLSI layout architectures such as gate matrix layout, folding of programmable logic arrays, and Weinberger arrays. These include a variety of mostly independently investigated graph problems such as augmentation of a given graph to an interval graph with small clique size, node search of graphs, matching problems with side constraints, and other. We discuss implications of graph theoretic results for the VLSI layout problems and survey new research directions. New results presented include NP-hardness of gate matrix layout on chordal graphs, efficient algorithms for trees, cographs, and certain chordal graphs, Lagrangean relaxation and approximation algorithms based on on-line interval graph augmentation.Key wordslinear VLSI layout architecturesgate matrix layoutPLA foldinginterval graph augmentationgraph searchpath widthvertex separationcomplexityapproximation algorithmsmatching with side constraints

Finding depth-first-search (DFS) trees of graphs is one of the fundamental problems in graph theory which has many practical applications. However, there exists no NC parallel algorithm for this problem in general graphs. NC parallel algorithms for finding DFS trees for the interval graphs and circular-arcs graphs are presented. These algorithms take O(log n) time using kappa n processors on the EREW model, where kappa is the number of cliques in a graph of n nodes. >

For a finite undirected graph G = (V ,E ) and positive integer k *** 1, an edge set M *** E is a distance-k matching if the mutual distance of edges in M is at least k in G . For k = 1, this gives the usual notion of matching in graphs, and for general k *** 1, distance-k matchings were called k-separated matchings by Stockmeyer and Vazirani. The special case k = 2 has been studied under the names induced matching (i.e., a matching which forms an induced subgraph in G ) by Cameron and strong matching by Golumbic and Laskar in various papers. Finding a maximum induced matching is $\mathbb{NP}$-complete even on very restricted bipartite graphs but for k = 2, it can be done efficiently on various classes of graphs such as chordal graphs, based on the fact that an induced matching in G corresponds to an independent vertex set in the square L (G )2 of the line graph L (G ) of G which, by a result of Cameron, is chordal for any chordal graph G . We show that, unlike for k = 2, for a chordal graph G , L (G )3 is not necessarily chordal, and finding a maximum distance-3 matching remains $\mathbb{NP}$-complete on chordal graphs. For strongly chordal graphs and interval graphs, however, the maximum distance-3 matching problem can be solved in polynomial time. Moreover, we obtain various new results for induced matchings.

Chordal graphs are the intersection graphs of subtrees of a tree, while interval graphs of subpaths of a path. Undirected path graphs, directed path graphs and rooted directed path graphs are intermediate graph classes, defined, respectively, as the intersection graphs of paths of a tree, of directed paths of an oriented tree, and of directed paths of an out branching. All of these path graphs have vertex leafage 2. Dominating Set, Connected Dominating Set, and Steiner tree problems are ‐hard parameterized by the size of the solution on chordal graphs, ‐complete on undirected path graphs, and polynomial‐time solvable on rooted directed path graphs, and hence also on interval graphs. We further investigate the (parameterized) complexity of all these problems when constrained to chordal graphs, taking the vertex leafage and the aforementioned classes into consideration. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are on chordal graphs when parameterized by the size of the solution plus the vertex leafage, and that Weighted Connected Dominating Set is polynomial‐time solvable on strongly chordal graphs. We also introduce a new subclass of undirected path graphs, which we call in–out rooted directed path graphs, as the intersection graphs of directed paths of an in–out branching. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are solvable in polynomial time on this class, generalizing the polynomiality for rooted directed path graphs proved by Booth and Johnson (SIAM J. Comput. 11 (1982), 191‐199.) and by White et al. (Networks 15 (1985), 109‐124.).

The subject of the research is one of the ways of updating modern training systems for solving problems of graph theory, namely, automatic generation of graphs. This approach will reduce the load on the training system database and generate tasks for the user in real-time without updating the bank of tasks. In the course of the work, the advantages and disadvantages of this approach were identified. The most suitable method for the implementation of the research was chosen to represent graphs in electronic computers. The requirements for generated graphs and possible ways of implementing these requirements are identified and substantiated. Namely: in the implemented program, simple connected undirected graphs will be generated. We considered an important detail in working with graphs — graph traversal using the “Depth (width) search” algorithm, which in this task is used to check the graph for connectivity. The result of the work is presented — a software implementation of the graph generation algorithm in the C# programming language. In it, graphs are represented by an adjacency list, generated randomly, and checked for connectivity using the DFS (Depth First Search) function. DFS is a software implementation of the Depth First Search algorithm.

The graph-search algorithms developed between 60s and 80s were widely used in many fields, from robotics to video games. The A* algorithm shall be mentioned between some of the most important solutions explicitly oriented to motion-robotics, improving the logic of graph search with heuristic principles inside the loop. Nevertheless, one of the most important drawbacks of the A* algorithm resides in the heading constraints connected with the grid characteristics. Different solutions were developed in the last years to cope with this problem, based on post-processing algorithms or on improvements of the graph-search algorithm itself. A very important one is Theta* that refines the graph search allowing to obtain paths with any heading. In the last two years, the Flight Mechanics Research Group of Politecnico di Torino studied and implemented different path planning algorithms. A Matlab based planning tool was developed, collecting four separate approaches: geometric predefined trajectories, manual waypoint definition, automatic waypoint distribution (i.e. optimizing camera payload capabilities) and a comprehensive A*-based algorithm used to generate paths, minimizing risk of collision with orographic obstacles. The tool named PCube exploits Digital Elevation Maps (DEMs) to assess the risk maps and it can be used to generate waypoint sequences for UAVs autopilots. In order to improve the A*-based algorithm, the solution is extended to tri-dimensional environments implementing a more effective graph search (based on Theta*). In this paper the application of basic Theta* to tri-dimensional path planning will be presented. Particularly, the algorithm is applied to orographic obstacles and in urban environments, to evaluate the solution for different kinds of obstacles. Finally, a comparison with the A* algorithm will be introduced as a metric of the algorithm performances.

This review identifies graph traversal methods as fundamental steps in network analysis that facilitate easy examination of relations within the network. In this review, we plan to discuss multiple graph traversal approaches like depth first search (DFS), breadth first search (BFS), their varieties and indicate how these techniques can be used in practice when analyzing networks. The following is a review methodology covering 10 articles for years 2019 – 2024. The study responds to the question of how to select algorithms for various types of networks with reference to their computational cost and speed. Thus, the authors describe the advantages and drawbacks of the reported techniques, and the topics serve as recommendations for the further advancement of technology. In this respect, the present review also serves to help to close the divide between theory and practice.

Author(s): Lueker, George S.; Booth, Kellogg S. | Abstract: A graph is an interval graph if and only if each of its vertices can be associated with an interval on the real line in such a way that two vertices are adjacent in the graph exactly when the corresponding intervals have a nonempty intersection. An efficient algorithm for testing isomorphism of interval graphs is implemented using a data structure called a PQ-tree. The algorithm runs in 0(n + e) steps for graphs having n vertices and e edges. It is shown that for a somewhat larger class of graphs, namely the chordal graphs, isomorphism is as hard as for general graphs.

Many graph search algorithms use a vertex labeling to compute an ordering of the vertices. We examine such algorithms which compute a peo (perfect elimination ordering) of a chordal graph and corresponding algorithms which compute an meo (minimal elimination ordering) of a non-chordal graph, an ordering used to compute a minimal triangulation of the input graph. We express all known peo-computing search algorithms as instances of a generic algorithm called MLS (maximal label search) and generalize Algorithm MLS into CompMLS, which can compute any peo. We then extend these algorithms to versions which compute an meo and likewise generalize all known meo-computing search algorithms. We show that not all minimal triangulations can be computed by such a graph search, and, more surprisingly, that all these search algorithms compute the same set of minimal triangulations, even though the computed meos are different. Finally, we present a complexity analysis of these algorithms. An extended abstract of part of this paper was published in WG 2005 .

Corneil, Olariu, and Stewart [SODA 1998; SIAM Journal on Discrete Mathematics 2009] presented a recognition algorithm for interval graphs by six graph searches. Li and Wu [Discrete Mathematics \& Theoretical Computer Science 2014] simplified it to only four. The great simplicity of the latter algorithm is however eclipsed by the complicated and long proofs. The main purpose of this paper is to present a new and significantly short proof for Li and Wu's algorithm, as well as a simpler implementation. We also give a self-contained simpler interpretation of the recognition algorithm of Corneil [Discrete Applied Mathematics 2004] for unit interval graphs, based on three sweeps of graph searches. Moreover, we show that two sweeps are already sufficient. Toward the proofs of the main results, we make several new structural observations that might be of independent interests.