An improved linear placement algorithm using node compaction

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

A graph is called Hamiltonian connected if it contains a Hamiltonian path between any two distinct vertices. In the past, we proved the Hamiltonian path and cycle problems for general supergrid graphs to be NP-complete. However, they are still open for solid supergrid graphs. In this paper, first we will verify the Hamiltonian cycle property of C-shaped supergrid graphs, which are a special case of solid supergrid graphs. Next, we show that C-shaped supergrid graphs are Hamiltonian connected except in a few conditions. For these excluding conditions of Hamiltonian connectivity, we compute their longest paths. Then, we design a linear-time algorithm to solve the longest path problem in these graphs. The Hamiltonian connectivity of C-shaped supergrid graphs can be applied to compute the optimal stitching trace of computer embroidery machines, and construct the minimum printing trace of 3D printers with a C-like component being printed.

We consider the following problem. Let G=(V, E) be an undirected planar graph and let s, t∈V, s≠t. The problem is to find a set of pairwise edge-disjoint paths, each connecting s with t, with maximum cardinality. In other words, the problem is to find a maximum unit flow from s to t. The fastest algorithm in the literature has running time O(|V|log|V|)(¦V¦ log ¦V¦). In this paper now, we give a linear time algorithm.KeywordsPlanar GraphLinear TimeUnit FlowOuter FaceSearch PathThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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 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

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