32 research outputs found

    Parameterized Graph Modification Beyond the Natural Parameter

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    Parameterized Graph Modification Beyond the Natural Parameter

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    A Polynomial-Time Algorithm for MCS Partial Search Order on Chordal Graphs

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    We study the partial search order problem (PSOP) proposed recently by Scheffler [WG 2022]. Given a graph GG together with a partial order over the vertices of GG, this problem determines if there is an S\mathcal{S}-ordering that is consistent with the given partial order, where S\mathcal{S} is a graph search paradigm like BFS, DFS, etc. This problem naturally generalizes the end-vertex problem which has received much attention over the past few years. It also generalizes the so-called F{\mathcal{F}}-tree recognition problem which has just been studied in the literature recently. Our main contribution is a polynomial-time dynamic programming algorithm for the PSOP on chordal graphs with respect to the maximum cardinality search (MCS). This resolves one of the most intriguing open questions left in the work of Sheffler [WG 2022]. To obtain our result, we propose the notion of layer structure and study numerous related structural properties which might be of independent interest.Comment: 12 page

    EUROCOMB 21 Book of extended abstracts

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    Graph classes and forbidden patterns on three vertices

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    This paper deals with graph classes characterization and recognition. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately this strategy usually does not lead to an efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques based on some interesting orderings of the nodes, such as the ones given by traversals. We study specifically graph classes that have an ordering avoiding some ordered structures. More precisely, we consider what we call patterns on three nodes, and the recognition complexity of the associated classes. In this domain, there are two key previous works. Damashke started the study of the classes defined by forbidden patterns, a set that contains interval, chordal and bipartite graphs among others. On the algorithmic side, Hell, Mohar and Rafiey proved that any class defined by a set of forbidden patterns can be recognized in polynomial time. We improve on these two works, by characterizing systematically all the classes defined sets of forbidden patterns (on three nodes), and proving that among the 23 different classes (up to complementation) that we find, 21 can actually be recognized in linear time. Beyond this result, we consider that this type of characterization is very useful, leads to a rich structure of classes, and generates a lot of open questions worth investigating.Comment: Third version version. 38 page

    Linear Time LexDFS on Chordal Graphs

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    Lexicographic Depth First Search (LexDFS) is a special variant of a Depth First Search (DFS), which was introduced by Corneil and Krueger in 2008. While this search has been used in various applications, in contrast to other graph searches, no general linear time implementation is known to date. In 2014, K\"ohler and Mouatadid achieved linear running time to compute some special LexDFS orders for cocomparability graphs. In this paper, we present a linear time implementation of LexDFS for chordal graphs. Our algorithm is able to find any LexDFS order for this graph class. To the best of our knowledge this is the first unrestricted linear time implementation of LexDFS on a non-trivial graph class. In the algorithm we use a search tree computed by Lexicographic Breadth First Search (LexBFS)

    Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)

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    International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference “Algebras, graphs and ordered set” (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Maurice’s many scientific interests:• Lattices and ordered sets• Combinatorics and graph theory• Set theory and theory of relations• Universal algebra and multiple valued logic• Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..

    距離独立集合問題および誘導マッチング問題に対する近似アルゴリズム

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    This thesis deals with the following two problems, the Maximum Distance-d Independent Set problem (MaxDdIS for short) and the Maximum Induced Matching problem (MaxIM for short), where d ≥ 3. We design some approximation algorithms to solve MaxDdIS and MaxIM. (1) We first study MaxDdIS. Our main results for MaxDdIS are as follows: (i) It is NP-hard to approximate MaxD3IS on 3-regular graphs within 1.00105 unless P=NP. (ii) For every fixed integers d ≥ 3 and r ≥ 3, MaxDdIS on r-regular graphs is APX-hard, and show the inapproximability of MaxDdIS on r-regular graphs. (iii) We design polynomial-time O(rd-1)-approximation and O(rd-2/d)- approximation algorithms for MaxDdIS on r-regular graphs. (iv) We sharpen the above O(rd-2/d)-approximation algorithms when restricted to d = r = 3, and give a polynomial-time 2-approximation algorithm for MaxD3IS on cubic graphs. (v) Furthermore, we design a polynomial-time 1.875-approximation algorithm for MaxD3IS on cubic graphs. (vi) Finally, we consider planar graphs and obtain that MaxDdIS admits a polynomial-time approximation scheme (PTAS) for planar graphs. (2) We then investigate MaxIM on r-regular graphs. For subclasses of r-regular graphs, several better approximation algorithms are known. The previously known best approximation ratios for MaxIM on C5-free r-regular graphs and C3, C5 -free r-regular graphs are (3r/4 - 1/8 + 3/16r - 8) and (0.7084r + 0.425), respectively. We design a (2r/3 + 1/3) -approximation algorithm, whose approximation ratio is strictly smaller/better than the previous one for C5-free r-regular graphs when r ≥ 6, and for {C3, C5 }-free r-regular graphs when r ≥ 3.九州工業大学博士学位論文 学位記番号:情工博甲第339号 学位授与年月日:平成31年3月25日1 Introduction|2 Preliminaries|3 Maximum Distance-d Independent Set problem|4 Maximum Induced Matching Problem|5 Conclusion九州工業大学平成30年

    Graph Searches and Their End Vertices

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    Graph search, the process of visiting vertices in a graph in a specific order, has demonstrated magical powers in many important algorithms. But a systematic study was only initiated by Corneil et al.~a decade ago, and only by then we started to realize how little we understand it. Even the apparently na\"{i}ve question "which vertex can be the last visited by a graph search algorithm," known as the end vertex problem, turns out to be quite elusive. We give a full picture of all maximum cardinality searches on chordal graphs, which implies a polynomial-time algorithm for the end vertex problem of maximum cardinality search. It is complemented by a proof of NP-completeness of the same problem on weakly chordal graphs. We also show linear-time algorithms for deciding end vertices of breadth-first searches on interval graphs, and end vertices of lexicographic depth-first searches on chordal graphs. Finally, we present 2nnO(1)2^n\cdot n^{O(1)}-time algorithms for deciding the end vertices of breadth-first searches, depth-first searches, maximum cardinality searches, and maximum neighborhood searches on general graphs
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