Three-coloring triangle-free planar graphs in linear time

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

The overwhelming need to improve the quality of complex data structures in healthcare is more important than ever. Although data quality has been the point of interest in many studies, none of them has focused on the development of quantitative and explainable methods for data imputation. In this work, we propose a "smart" imputation workflow to address missing data across complex data structures in the context of in silico clinical trials. AI algorithms were utilized to produce high-quality virtual patient profiles. A search algorithm was then developed to extract the best virtual patient profiles through the definition of a profile matching score (PMS). A case study was conducted, where the real dataset was randomly contaminated with multiple missing values (e.g., 10 to 50%). In total, 10000 virtual patient profiles with less than 0.02 Kullback-Leibler (KL) divergence were produced to estimate the PMS distribution. The best generator achieved the lowest average squared absolute difference (0.4) and average correlation difference (0.02) with the real dataset highlighting its increased effectiveness for data imputation across complex clinical data structures.

Given a plane graph it is known how to compute the union of non-crossing shortest paths. These algorithms do not allow neither to list each single shortest path nor to compute length of shortest paths. Given the union of non-crossing shortest paths, we introduce the concept of shortcuts that allows us to establish whether a path is a shortest path by checking local properties on faces of the graph. By using shortcuts we can compute the length of each shortest path, given their union, in total linear time, and we can list each shortest path p in O(max{ℓ,ℓloglogkℓ}), where ℓ is the number of edges in p and k the number of shortest paths.

Let G be a finite undirected graph with edge set E. An edge set E′⊆E is an induced matching in G if the pairwise distance of the edges of E′ in G is at least two; E′ is dominating in G if every edge e∈E∖E′ intersects some edge in E′. The Dominating Induced Matching Problem (DIM, for short) asks for the existence of an induced matching E′ which is also dominating in G; this problem is also known as the Efficient Edge Domination Problem. The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is ${\mathbb{NP}}$ -complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three. However, its complexity was open for P k -free graphs for any k≥5; P k denotes a chordless path with k vertices and k−1 edges. We show in this paper that the weighted DIM problem is solvable in linear time for P 7-free graphs in a robust way.

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

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 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.

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.

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.

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.

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