Polynomial Algorithms for Hamiltonian Cycle in Cocomparability Graphs

Hamiltonian cycle is polynomial on cocomparability graphs

Graph theory is an important topic in discrete mathematics. It is particularly interesting because it has a wide range of applications. Among the main problems in graph theory, we shall mention the following ones: graph coloring and the Hamiltonian circuit problem. Chapter 1 presents basic definitions of graph theory, such as graph coloring, graph coloring with color-classes of bounded size b, and Hamiltonian circuits and paths. We also present online algorithms and online coloring. Chapter 2 starts with some general remarks about online graph covering with sets of bounded sizes (such as online bounded coloring): we give a simple method for transforming an online covering algorithm into an online bounded covering algorithm, and to derive the performance ratio of the bounded algorithm from the performance ratio of the unbounded algorithm. As will be shown in later chapters, this method often leads to optimal results. Furthermore, some basic preliminary results on online graph covering with sets of bounded size are given: for every graph, the performance ratio is bounded above by 1/2 + b/2 and for b = 2, this bound is optimal. In the second part, online coloring of co-interval graphs is studied. Based on two industrial applications, two different versions of this problem are discussed. In the case where the intervals are presented in increasing order of their left ends, we show that the performance ratio is 1 in the unbounded case and 2 - 1/b in the bounded case. In the case where the intervals may be presented in any order, we show that the performance ratio is at most 3 in the bounded case. Chapter 3 deals with online coloring of permutation and comparability graphs. First, we give a tight analysis of the First-Fit algorithm on bipartite permutation graphs and we show that its performance ratio is O(√n), even for some simple presentation orders. For both classes of graphs, we show that the performance ratio is bounded above by (χ+1)/2 in the unbounded case and that the performance ratio of First-Fit is equal to 1/2 + b/2 in the bounded case. In the second part of this chapter, we study cocoloring of permutation graphs. We show that the performance ratio is n/4 + 1/2 and we give better bounds in some more restricted cocoloring problems. Chapter 4 deals with an application of online coloring: the online Track Assignment Problem. Depending on the assumptions that are made, the Track Assignment Problem can be reduced to coloring permutation or overlap graphs online. We show that when a permutation graph is presented on a latticial plane, from west to east, then the performance ratio is exactly 2 - (min{b,k})-1, where k is the best known upper bound on the bounded chromatic number. We also show that, when a permutation graph is presented on a latticial plan, starting from the origin and growing, simultaneously or not, towards west and east, then the performance ratio is exactly 2 - 1/χ. We also show that online coloring overlap graphs does not have a performance ratio bounded by a constant, even if the overlap graph is bipartite and presented in increasing order of the intervals left ends. In this special case, we show that First-Fit has a tight performance ratio of O(√n). We consider coloring overlap graphs online where the intervals have a bounded size between 1 and a given number M. In this case, we show that the performance ratio can be bounded above by 2√M if M ≤ M0, and by log M (⎡log M / log log M⎤ + 1) if M > M0, M0 being defined by the equation 2√M0 = 3 log(M0). For large values of M, the ratio is O(log2 M / log log M). Chapter 5 is about online coloring of trees, forests and split-graphs. For trees, we show that the performance ratio of online coloring is exactly ½log2(2n) in the unbouded case and at most 1 + ⎣log2(b)⎦/χb in the bounded case. For split-graphs, we show that the performance ratio of online coloring is exactly 1 + 1/χ in the unbounded case and is at most 2 + 1/χb + 3/b in the bounded case. In Chapter 6, we present a class of digraphs: the quasi-adjoint graphs. These are a super class of both the graphs used for a DNA sequencing algorithm in (Blazewicz, Kasprzak, "Computational complexity of isothermic DNA sequencing by hybridization", 2006) and the adjoints. A polynomial recognition algorithm in O(n3), as well as a polynomial algorithm in O(n2 + m2) for finding a Hamiltonian circuit in quasi-adjoint graphs are given. Furthermore, some results about related problems such as finding a Eulerian circuit while respecting some forbidden transitions (a sequence of two consecutive arcs) are discussed.

For graph $G(V,E)$, a minimum path cover (MPC) is a minimum cardinality set of vertex disjoint paths that cover $V$ (i.e., every vertex of $G$ is in exactly one path in the cover). This problem is a natural generalization of the Hamiltonian path problem. Cocomparability graphs (the complements of graphs that have an acyclic transitive orientation of their edge sets) are a well studied subfamily of perfect graphs that includes many popular families of graphs such as interval, permutation, and cographs. Furthermore, for every cocomparability graph $G$ and acyclic transitive orientation of the edges of $\overline{G}$ there is a corresponding poset $P_G$; it is easy to see that an MPC of $G$ is a linear extension of $P_G$ that minimizes the bump number of $P_G$. Although there are directly graph-theoretical MPC algorithms (i.e., algorithms that do not rely on poset formulations) for various subfamilies of cocomparability graphs, notably interval graphs, until now all MPC algorithms for cocomparability graphs themselves have been based on the bump number algorithms for posets. In this paper we present the first directly graph-theoretical MPC algorithm for cocomparability graphs; this algorithm is based on two consecutive graph searches followed by a certifying algorithm. Surprisingly, except for a lexicographic depth first search (LDFS) preprocessing step, this algorithm is identical to the corresponding algorithm for interval graphs. The running time of the algorithm is $O({\rm min}(n^2, n + {\rm mloglogn}))$, with the nonlinearity coming from LDFS.

A graph $G$ is a cocomparability graph if there exists an acyclic transitive orientation of the edges of its complement graph $\overline{G}$. LBFS$^{+}$ is a variant of the generic Lexicographic Breadth First Search (LBFS), which uses a specific tie-breaking mechanism. Starting with some ordering $\sigma_{0}$ of $G$, let $\{\sigma_{i}\}_{i\geq 1}$ be the sequence of orderings such that $\sigma_{i}=$LBFS$^{+}(G, \sigma_{i-1})$. The LexCycle($G$) is defined as the maximum length of a cycle of vertex orderings of $G$ obtained via such a sequence of LBFS$^{+}$ sweeps. Dusart and Habib conjectured in 2017 that LexCycle($G$)=2 if $G$ is a cocomparability graph and proved it holds for interval graphs. In this paper, we show that LexCycle($G$)=2 if $G$ is a $\overline{P_{2}\cup P_{3}}$-free cocomparability graph, where a $\overline{P_{2}\cup P_{3}}$ is the graph whose complement is the disjoint union of $P_{2}$ and $P_{3}$. As corollaries, it's applicable for diamond-free cocomparability graphs, cocomparability graphs with girth at least 4, as well as interval graphs.Comment: 11 pages, 9 figures

Summary of Research Our research efforts funded by ONR have produced excellent results in several different areas of graph theory and its applications. Hamiltonian Graphs: In Hamiltonian graphs our approach was twofold. On one hand we investigated sufficient degree and edge conditions for balanced bipartite graphs to be hamiltonian. On the other hand, we developed algorithms for finding Hamiltonian paths and cycles in permutation and cocomparability graphs. It may be noted that Hamiltonian cycle problem was well known open problem since 1985. We developed 0(n 2 ) algorithm for permutation graphs and 0{n z ) algorithm for cocomparability graphs. In addition, toughness properties of permutation graphs and cocomparability graphs were also investigated. These results contribute significantly to the understanding of Hamiltonian properties. Line Graphs and their generalizations: Line graphs provide a way of studying the graph by concentrating attention on edges without regard to vertices. We generalized the notion of fine graphs to super line graphs and obtained several results about their properties. Our approach studies fine graphs combinatorially, by looking at sets of edges of a given cardinality. Several interresting new parameters related to the notion of super fine graphs have been introduced and studied. This study contributes significantly to the generalizations of the line graph transformation.

A new LBFS-based algorithm for cocomparability graph recognition

The longest path problem is the problem of finding a path of maximum length in a graph. As a generalization of the Hamiltonian path problem, it is NP-complete on general graphs and, in fact, on every class of graphs that the Hamiltonian path problem is NP-complete. Polynomial solutions for the longest path problem have recently been proposed for weighted trees, Ptolemaic graphs, bipartite permutation graphs, interval graphs, and some small classes of graphs. Although the Hamiltonian path problem on cocomparability graphs was proved to be polynomial almost two decades ago, the complexity status of the longest path problem on cocomparability graphs has remained open; actually, the complexity status of the problem has remained open even on the smaller class of permutation graphs. In this paper, we present a polynomial-time algorithm for solving the longest path problem on the class of cocomparability graphs. Our result resolves the open question for the complexity of the problem on such graphs, and since cocomparability graphs form a superclass of both interval and permutation graphs, extends the polynomial solution of the longest path problem on interval graphs and provides polynomial solution to the class of permutation graphs.

The longest path problem is the problem of finding a path of maximum length in a graph. As a generalization of the Hamiltonian path problem, it is NP-complete on general graphs and, in fact, on every class of graphs that the Hamiltonian path problem is NP-complete. Polynomial solutions for the longest path problem have recently been proposed for weighted trees, ptolemaic graphs, bipartite permutation graphs, interval graphs, and some small classes of graphs. Although the Hamiltonian path problem on cocomparability graphs was proved to be polynomial almost two decades ago[9], the complexity status of the longest path problem on cocomparability graphs has remained open until now; actually, the complexity status of the problem has remained open even on the smaller class of permutation graphs. In this paper, we present a polynomial-time algorithm for solving the longest path problem on the class of cocomparability graphs. Our result resolves the open question for the complexity of the problem on such graphs, and since cocomparability graphs form a superclass of both interval and permutation graphs, extends the polynomial solution of the longest path problem on interval graphs[18] and provides polynomial solution to the class of permutation graphs.KeywordsLongest path problemcocomparability graphspermutation graphspolynomial algorithmcomplexity

Hamiltonian Path/Cycle are well known NP-complete problems on general graphs, but their complexity status for permutation graphs has been an open question in algorithmic graph theory for many years. In this paper, we prove that theHamiltonian Path problem is solvable in polynomial time even for the larger class of cocomparability graphs. Our result is based on a nice relationship between Hamiltonian paths and the bump number of partial orders. As another consequence we get a new interpretation of the bump number in terms of path partitions, leading to polynomial time solutions of theHamiltonian Path/Cycle Completion problems in cocomparability graphs.

A linear time algorithm to compute a maximum weighted independent set on cocomparability graphs

An NC algorithm for the clique cover problem in cocomparability graphs and its application

In this chapter, the concepts of Hamiltonian paths and Hamiltonian cycles are discussed. In the first section, the history of Hamiltonian graphs is described, and then some concepts such as Hamiltonian paths, Hamiltonian cycles, traceable graphs, and Hamiltonian graphs are defined. Also some most known Hamiltonian graph problems such as travelling salesman problem (TSP), Kirkman’s cell of a bee, Icosian game, and knight’s tour problem are presented. In addition, necessary and (or) sufficient conditions for existence of a Hamiltonian cycle are investigated. Furthermore, in order to solve Hamiltonian cycle problems, some algorithms are introduced in the last section.

The Hamiltonian properties of supergrid graphs

1-Tough cocomparability graphs are hamiltonian

A unit cube in k dimensions (k-cube) is defined as the Cartesian product R 1×R 2× ⋯ ×R k where R i (for 1 ≤ i ≤ k) is a closed interval of the form [a i ,a i + 1] on the real line. A graph G on n nodes is said to be representable as the intersection of k-cubes (cube representation in k dimensions) if each vertex of G can be mapped to a k-cube such that two vertices are adjacent in G if and only if their corresponding k-cubes have a non-empty intersection. The cubicity of G denoted as cub(G) is the minimum k for which G can be represented as the intersection of k-cubes.An interesting aspect about cubicity is that many problems known to be NP-complete for general graphs have polynomial time deterministic algorithms or have good approximation ratios in graphs of low cubicity. In most of these algorithms, computing a low dimensional cube representation of the given graph is usually the first step.We give an O(bw·n) algorithm to compute the cube representation of a general graph G in bw + 1 dimensions given a bandwidth ordering of the vertices of G, where bw is the bandwidth of G. As a consequence, we get O(Δ) upper bounds on the cubicity of many well-known graph classes such as AT-free graphs, circular-arc graphs and cocomparability graphs which have O(Δ) bandwidth. Thus we have: 1 cub(G) ≤ 3Δ− 1, if G is an AT-free graph. 1 cub(G) ≤ 2Δ + 1, if G is a circular-arc graph. 1 cub(G) ≤ 2Δ, if G is a cocomparability graph. Also for these graph classes, there are constant factor approximation algorithms for bandwidth computation that generate orderings of vertices with O(Δ) width. We can thus generate the cube representation of such graphs in O(Δ) dimensions in polynomial time.KeywordsCubicitybandwidthintersection graphsAT-free graphscircular-arc graphscocomparability graphs