Search Numbers in Networks with Special Topologies

In this paper, we study the problem of computing the minimum number of searchers who can capture an intruder hiding in a graph. We propose a linear time algorithm for computing the vertex separation and the optimal layout for a unicyclic graph. The best algorithm known so far is given by Ellis et al. (2004) and needs O(n log n) time, where n is the number of vertices in the graph. By a linear-time transformation, we can compute the search number and the optimal search strategy for a unicyclic graph in linear time. We show how to compute the search number for a k-ary cycle-disjoint graph. We also present some results on approximation algorithms.

A k-colouring of a graph G is an assignment of k different colours to the vertices of G such that adjacent vertices receive different colours. The minimum cardinality k for which G has a k-colouring is called the chromatic number of G and is denoted by χ(G). Its computation is NP-hard. It is not difficult to colour the vertices of a graph in linear time using at most Δ(G) + 1 colours, where Δ(G) is the maximum vertex degree of a given graph G. Moreover, the classical theorem of Brooks states that χ(G) ≤ Δ(G) unless G is a complete graph or an odd cycle. We will strengthen the upper bound of Δ + 1 by considering additional parameters in terms of vertex degrees and the clique number. These new bounds will be accompanied by polynomial time algorithms attaining these improved bounds. †Dedicated to H. Th. Jongen on the occasion of his 60th birthday.

Linear placement has several applications in the design of very large scale integrated (VLSI) circuits. Several authors have recently shown that node clustering or compaction enhances the performance of iterative algorithms. This paper describes a technique to extract clusters by means of an algorithm that identifies a maximum weight matching in a path graph in linear time. The resulting linear placement algorithm (CLP) combines node compactions and iterative improvement in an efficient and effective approach. The benefit of compaction is quite visible. Comparisons of the results of CLP with and without compaction on several test cases show the dramatic effect of compaction on the final solution. The approach of CLP is an efficient improvement of the linear placement algorithm of Saab and Chen [1994]. On many instances with known optimal solutions, CLP also achieves an optimal solution. The dependency of the results of CLP on the initial placement is minimal, since solutions found by CLP using many different starts do not significantly differ from each other. Under suitable assumption, CLP can be shown to run in O(nP), where n and P are the number of nodes and the number of pins of the input circuit, respectively.

Labelled packing functions in graphs

A linear time algorithm for 7-[3]coloring triangle-free hexagonal graphs

An implicit representation of chordal comparability graphs in linear time

Dominating induced matching in some subclasses of bipartite graphs

In this paper we show how taking advantage of autonomous agents' capability to maintain parallel interactions with others, and incorporating it into the cooperative economic search model results in a new search strategy which outperforms current strategies in use. As a framework for our analysis we use the electronic marketplace, where buyer agents have the incentive to search cooperatively. The new search technique is quite intuitive, however its analysis and the process of extracting the optimal search strategy are associated with several significant complexities. These difficulties are derived mainly from the unbounded search space and simultaneous dual affects of decisions taken along the search. We provide a comprehensive analysis of the model, highlighting, demonstrating and proving important characteristics of the optimal search strategy. Consequently, we manage to come up with an efficient modular algorithm for extracting the optimal cooperative search strategy for any given environment. A computational based comparative illustration of the system performance using the new search technique versus the traditional methods is given, emphasizing the main differences in the optimal strategy's structure and the advantage of using the proposed model.

Grötzsch's theorem states that every triangle-free planar graph is 3-colorable, and several relatively simple proofs of this fact were provided by Thomassen and other authors. It is easy to convert these proofs into quadratic-time algorithms to find a 3-coloring, but it is not clear how to find such a coloring in linear time (Kowalik used a nontrivial data structure to construct anO(nlogn) algorithm).We design a linear-time algorithm to find a 3-coloring of a given triangle-free planar graph. The algorithm avoids using any complex data structures, which makes it easy to implement. As a by-product, we give a yet simpler proof of Grötzsch's theorem.

Grotzsch's theorem states that every triangle-free planar graph is 3-colorable, and several relatively simple proofs of this fact were provided by Thomassen and other authors. It is easy to convert these proofs into quadratic-time algorithms to find a 3-coloring, but it is not clear how to find such a coloring in linear time (Kowalik used a nontrivial data structure to construct an O(n log n) algorithm).We design a linear-time algorithm to find a 3-coloring of a given triangle-free planar graph. The algorithm avoids using any complex data structures, which makes it easy to implement. As a by-product we give a yet simpler proof of Grotzsch's theorem.

Approximate tree decompositions of planar graphs in linear time

Rainbow colouring of split graphs

Construction of a simple elimination scheme for a chordal comparability graph in linear time

Semantic understanding is an essential research issue for many applications, such as social network analysis, collective intelligence and content computing, which tells the inner meaning of language form. Recently, Abstract Meaning Representation (AMR) is attracted by many researchers for its semantic representation ability on an entire sentence. However, due to the non-projectivity and reentrancy properties of AMR graphs, they lose some important semantic information in parsing from sentences. In this paper, we propose a general AMR parsing model which utilizes a two-stack-based transition algorithm for both Chinese and English datasets. It can incrementally parse sentences to AMR graphs in linear time. Experimental results demonstrate that it is superior in recovering reentrancy and handling arcs while is competitive with other transition-based neural network models on both English and Chinese datasets.

Almost diameter of a house-hole-free graph in linear time via LexBFS