Linear Time LexDFS on Cocomparability Graphs

Lexicographic depth first search (LexDFS) is a graph search protocol which has already proved to be a powerful tool on cocomparability graphs. Cocomparability graphs have been well studied by investigating their complements (comparability graphs) and their corresponding posets. Recently however LexDFS has led to a number of elegant polynomial and near linear time algorithms on cocomparability graphs when used as a preprocessing step [2, 3, 11]. The nonlinear runtime of some of these results is a consequence of complexity of this preprocessing step. We present the first linear time algorithm to compute a LexDFS cocomparability ordering, therefore answering a problem raised in [2] and helping achieve the first linear time algorithms for the minimum path cover problem, and thus the Hamilton path problem, the maximum independent set problem and the minimum clique cover for this graph family

Linear Time LexDFS on Chordal Graphs

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)

(Quasi-)linear time algorithm to compute LexDFS, LexUP and LexDown orderings

We consider the three graph search algorithm LexDFS, LexUP and LexDOWN. We show that LexUP orderings can be computed in linear time by an algorithm similar to the one which compute LexBFS. Furthermore, LexDOWN orderings and LexDFS orderings can be computed in time $\left(n+m\log m\right)$ where $n$ is the number of vertices and $m$ the number of edges

Counting Independent Sets in Cocomparability Graphs

We show that the number of independent sets in cocomparability graphs can be counted in linear time, as can counting cliques in comparability graphs. By contrast, counting cliques in cocomparability graphs and counting independent sets in comparability graphs are #P-complete. We extend these results to counting maximal cliques and independent sets. We also consider the fixed-parameter versions of counting cliques and independent sets of given size k. Finally, we combine the results to show that both counting cliques and independent sets in permutation graphs are in linear time

Graphs with at most two moplexes

A moplex is a natural graph structure that arises when lifting Dirac's classical theorem from chordal graphs to general graphs. However, while every non-complete graph has at least two moplexes, little is known about structural properties of graphs with a bounded number of moplexes. The study of these graphs is motivated by the parallel between moplexes in general graphs and simplicial modules in chordal graphs: Unlike in the moplex setting, properties of chordal graphs with a bounded number of simplicial modules are well understood. For instance, chordal graphs having at most two simplicial modules are interval. In this work we initiate an investigation of $k$-moplex graphs, which are defined as graphs containing at most $k$ moplexes. Of particular interest is the smallest nontrivial case $k=2$, which forms a counterpart to the class of interval graphs. As our main structural result, we show that the class of connected $2$-moplex graphs is sandwiched between the classes of proper interval graphs and cocomparability graphs; moreover, both inclusions are tight for hereditary classes. From a complexity theoretic viewpoint, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. We develop new reductions that answer this question negatively for two prominent problems fitting this profile, namely Graph Isomorphism and Max-Cut. On the other hand, we prove that every connected $2$-moplex graph contains a Hamiltonian path, generalising the same property of connected proper interval graphs. Furthermore, for graphs with a higher number of moplexes, we lift the previously known result that graphs without asteroidal triples have at most two moplexes to the more general setting of larger asteroidal sets

Revisiting path-type covering and partitioning problems

This is a survey article which is at the initial stage. The author will appreciate to receive your comments and contributions to improve the quality of the article. The author's contact address is [email protected] problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering problem), covering the vertex set by independent sets (coloring problem), and covering the vertex set by paths or cycles. A similar concept which is partitioning problem is also equally important. Lately research in graph theory has produced unprecedented growth because of its various application in engineering and science. The covering and partitioning problem by paths itself have produced a sizable volume of literatures. The research on these problems is expanding in multiple directions and the volume of research papers is exploding. It is the time to simplify and unify the literature on different types of the covering and partitioning problems. The problems considered in this article are path cover problem, induced path cover problem, isometric path cover problem, path partition problem, induced path partition problem and isometric path partition problem. The objective of this article is to summarize the recent developments on these problems, classify their literatures and correlate the inter-relationship among the related concepts

Maximal cliques structure for cocomparability graphs and applications

Il s'agit d'une recherche sur les relations entre les graphes d'intervalles et les graphes de cocomparabilitéA cocomparability graph is a graph whose complement admits a transitive orientation. An interval graph is the intersection graph of a family of intervals on the real line. In this paper we investigate the relationships between interval and cocomparabil-ity graphs. This study is motivated by recent results [5, 13] that show that for some problems, the algorithm used on interval graphs can also be used with small modifications on cocomparability graphs. Many of these algorithms are based on graph searches that preserve cocomparability orderings. First we propose a characterization of cocomparability graphs via a lattice structure on the set of their maximal cliques. Using this characterization we can prove that every maximal interval subgraph of a cocomparability graph G is also a maximal chordal subgraph of G. Although the size of this lattice of maximal cliques can be exponential in the size of the graph, it can be used as a framework to design and prove algorithms on cocomparability graphs. In particular we show that a new graph search, namely Local Maximal Neighborhood Search (LocalMNS) leads to an O(n + mlogn) time algorithm to find a maximal interval subgraph of a cocomparability graph. Similarly we propose a linear time algorithm to compute all simplicial vertices in a cocomparability graph. In both cases we improve on the current state of knowledge

Influence of the tie-break rule on the end-vertex problem

International audienceEnd-vertices of a given graph search may have some nice properties, as for example it is well known that the last vertex of Lexicographic Breadth First Search (LBFS) in a chordal graph is simplicial, see Rose, Tarjan and Lueker 1976. Therefore it is interesting to consider if these vertices can be recognized in polynomial time or not, as first studied in Corneil, Köhler and Lanlignel 2010. A graph search is a mechanism for systematically visiting the vertices of a graph. At each step of a graph search, the key point is the choice of the next vertex to be explored. Graph searches only differ by this selection mechanism during which a tie-break rule is used. In this paper we study how the choice of the tie-break in case of equality during the search, for a given graph search including the classic ones such as BFS and DFS, can determine the complexity of the end-vertex problem. In particular we prove a counterintuitive NP-completeness result for Breadth First Search solving a problem raised in Corneil, Köhler and Lanlignel 2010

Efficient Algorithms on Cocomparability Graphs via Vertex Orderings

In this thesis, we give new structural and algorithmic results on cocomparability (ccp) graphs; particularly, by connecting structural properties of graph searching to those of ver- tex ordering characterizations and linear extensions of posets. To do so, we present new tools that we hope will prove useful on other applications – especially a new graph parti- tion refinement technique, and a new graph parameter that captures the linear structure exhibited by certain graph classes. The thesis consists of two main parts. In the first half, we give the first linear time algorithm to compute a LexDFS ordering of ccp graphs. Doing so, we overcome a bottleneck to achieving certifying linear time Hamilton path, max independent set and max matching algorithms on this graph class. Building on this result, we give a new simple and elegant algorithm to the classical minimum bump problem on posets. Lastly, we introduce a new graph invariant, LexCycle, which measures how linearly structured certain graph classes are, this linear structure has been a driving factor to a number of efficient algorithms. In the second half of the thesis, we lift our results to weighted ccp graphs. In particular, we give the first linear time robust algorithm for the weighted max independent set problem on this graph class; we then build on this result to give the fastest (O(mn) time) algorithm for the weighted induced matching problem on ccp graphs. In this latter work, we also give a unifying framework that captures graphs, characterized by forbidden orderings, which are closed under L2(·), the operation of computing the square of the line graph. We conclude by presenting structural properties on AT-free graphs, a superclass of ccp graphs, that we hope can be used to lift some of these results to this larger graph class.Ph.D