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

Rainbow colouring of split graphs

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.

In this paper, we consider the problem of finding the minimum number of searchers to sweep networks/graphs with special topological structures. Such a number is called the search number. We first study graphs, which contain only one cycle, and present a linear time algorithm to compute the vertex separation and the optimal layout of such graphs; by a linear-time transformation, we can find the search number of this kind of graphs in linear time. We also investigate graphs, in which every vertex lies on at most one cycle and each cycle contains at most three vertices of degree more than two, and we propose a linear time algorithm to compute their search number and optimal search strategy. We prove explicit formulas for the search number of the graphs obtained from complete k-ary trees by replacing vertices by cycles. We also present some results on approximation algorithms.

Labelled packing functions in graphs

An implicit representation of chordal comparability graphs in linear time

Dominating induced matching in some subclasses of bipartite graphs

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.

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

The Maximum Induced Matching (MIM) Problem asks for a largest set of pairwise vertex-disjoint edges in a graph which are pairwise of distance at least two. It is well-known that the MIM problem is NP-complete even on particular bipartite graphs and on line graphs. On the other hand, it is solvable in polynomial time for various classes of graphs (such as chordal, weakly chordal, interval, circular-arc graphs and others) since the MIM problem on graph G corresponds to the Maximum Independent Set problem on the square G *=L(G)2 of the line graph L(G) of G, and in some cases, G * is in the same graph class; for example, for chordal graphs G, G * is chordal. The construction of G *, however, requires ${\mathcal{O}}(m^{2})$time, where m is the number of edges in G. Is has been an open problem whether there is a linear-time algorithm for the MIM problem on chordal graphs. We give such an algorithm which is based on perfect elimination order and LexBFS.

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.

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