An enumeration kernel as defined by Creignou et al. (2017) [11] for a parameterized enumeration problem consists of an algorithm that transforms each instance into one whose size is bounded by the parameter plus a solution-lifting algorithm that efficiently enumerates all solutions from the set of the solutions of the kernel. We propose to consider two new versions of enumeration kernels by asking that the solutions of the original instance can be enumerated in polynomial time or with polynomial delay from the kernel solutions. Using the NP-hard Matching Cut problem parameterized by structural parameters such as the vertex cover number or the cyclomatic number of the input graph, we show that the new enumeration kernels present a useful notion of data reduction for enumeration problems which allows to compactly represent the set of feasible solutions.

In a graph, a matching cut is an edge cut that is a matching. Matching Cut, which is known to be NP-complete, is the problem of deciding whether or not a given graph G has a matching cut. In this paper we show that Matching Cut admits a quadratic-vertex kernel for the parameter distance to cluster and a linear-vertex kernel for the parameter distance to clique. We further provide an O^*(2^{dc(G)}) time and an O^*(2^{dc^-}(G)}) time FPT algorithm for Matching Cut, where dc(G) and dc^-(G) are the distance to cluster and distance to co-cluster, respectively. We also improve the running time of the best known branching algorithm to solve Matching Cut from O^*(1.4143^n) to O^*(1.3803^n). Moreover, we point out that, unless NP subseteq coNP/poly, Matching Cut does not admit a polynomial kernel when parameterized by treewidth.

We present space-efficient algorithms for computing cut vertices in a given graph with n vertices and m edges in linear time using $$O(n+\min \{m,n\log \log n\})$$ bits. With the same time and using $$O(n+m)$$ bits, we can compute the biconnected components of a graph. We use this result to show an algorithm for the recognition of (maximal) outerplanar graphs in $$O(n\log \log n)$$ time using O(n) bits.

A graph H is a square root of a graph G, or equivalently, G is the square of H, if G can be obtained from H by adding an edge between any two vertices in H that are of distance 2. The Square Root problem is that of deciding whether a given graph admits a square root. The problem of testing whether a graph admits a square root which belongs to some specified graph class $\mathcal {H}$ is called the $\mathcal {H}$ -Square Root problem. By showing boundedness of treewidth we prove that Square Root is polynomial-time solvable on some classes of graphs with small clique number and that $\mathcal {H}$ -Square Root is polynomial-time solvable when $\mathcal {H}$ is the class of cactuses.

A graph H is a square root of a graph G if G can be obtained from H by the addition of edges between any two vertices in H that are at distance 2 from each other. The Square Root problem is that of deciding whether a given graph admits a square root. We consider this problem for planar graphs in the context of the “distance from triviality” framework. For an integer k , a planar + k v graph (or k -apex graph) is a graph that can be made planar by the removal of at most k vertices. We prove that a generalization of Square Root , in which some edges are prescribed to be either in or out of any solution, has a kernel of size O ( k ) for planar + k v graphs, when parameterized by k . Our result is based on a new edge reduction rule which, as we shall also show, has a wider applicability for the Square Root problem.

Listing, generating or enumerating objects of specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. We study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality. We establish enumeration algorithms as well as lower and upper bounds for the maximum number of minimal connected dominating sets in such graphs. In particular, we present algorithms to enumerate all minimal connected dominating sets of chordal graphs in time O(1.7159^n), of split graphs in time O(1.3803^n), and of AT-free, strongly chordal, and distance-hereditary graphs in time O^*(3^{n/3}), where n is the number of vertices of the input graph. Our algorithms imply corresponding upper bounds for the number of minimal connected dominating sets for these graph classes.

In this note, we give a proof that several vertex ordering problems can be solved in O ∗(2 n ) time and O ∗(2 n ) space, or in O ∗(4 n ) time and polynomial space. The algorithms generalize algorithms for the Travelling Salesman Problem by Held and Karp (J. Soc. Ind. Appl. Math. 10:196–210, 1962) and Gurevich and Shelah (SIAM J. Comput. 16:486–502, 1987). We survey a number of vertex ordering problems to which the results apply.

We give experimental and theoretical results on the problem of computing the treewidth of a graph by exact exponential-time algorithms using exponential space or using only polynomial space. We first report on an implementation of a dynamic programming algorithm for computing the treewidth of a graph with running time O *(2 n ). This algorithm is based on the old dynamic programming method introduced by Held and Karp for the Traveling Salesman problem. We use some optimizations that do not affect the worst case running time but improve on the running time on actual instances and can be seen to be practical for small instances. We also consider the problem of computing Treewidth under the restriction that the space used is only polynomial and give a simple O *(4 n ) algorithm that requires polynomial space. We also show that with a more complicated algorithm using balanced separators, Treewidth can be computed in O *(2.9512 n ) time and polynomial space.

For more than 40 years, Branch & Reduce exponential-time backtracking algorithms have been among the most common tools used for finding exact solutions of NP-hard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worst-case running time bounds. Motivated by this, we use an approach, that we call “Measure & Conquer”, as an attempt to step beyond such limitations. The approach is based on the careful design of a nonstandard measure of the subproblem size; this measure is then used to lower bound the progress made by the algorithm at each branching step. The idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit, and hence lead to a significantly better worst-case time analysis. In order to show the potentialities of Measure & Conquer, we consider two well-studied NP-hard problems: minimum dominating set and maximum independent set. For the first problem, we consider the current best algorithm, and prove (thanks to a better measure) a much tighter running time bound for it. For the second problem, we describe a new, simple algorithm, and show that its running time is competitive with the current best time bounds, achieved with far more complicated algorithms (and standard analysis). Our examples show that a good choice of the measure, made in the very first stages of exact algorithms design, can have a tremendous impact on the running time bounds achievable.

The minimum dominating set problem remains NP-hard when restricted to any of the following graph classes: c -dense graphs, chordal graphs, 4-chordal graphs, weakly chordal graphs, and circle graphs. Developing and using a general approach, for each of these graph classes we present an exponential time algorithm solving the minimum dominating set problem faster than the best known algorithm for general graphs. Our algorithms have the following running time: O (1.4124 n ) for chordal graphs, O (1.4776 n ) for weakly chordal graphs, O (1.4845 n ) for 4-chordal graphs, O (1.4887 n ) for circle graphs, and O (1.2273 (1+√1−2 c ) n ) for c -dense graphs.

We show that the treewidth and the minimum fill-in of an n-vertex graph can be computed in time $\mathcal{O}(1.8899^n)$. Our results are based on combinatorial proofs that an n-vertex graph has $\mathcal{O}(1.7087^n)$ minimal separators and $\mathcal{O}(1.8135^n)$ potential maximal cliques. We also show that for the class of asteroidal triple–free graphs the running time of our algorithms can be reduced to $\mathcal{O}(1.4142^n)$.

A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation. We give linear-time certifying algorithms for recognition of interval graphs and permutation graphs, and for a few other related problems. Previous algorithms fail to provide supporting evidence when they claim that the input graph is not a member of the class. We show that our certificates of nonmembership can be authenticated in O(|V|) time.

This paper uses periodic matrix multiplication to improve the time complexities for a number of graph problems. The time for finding an asteroidal triple is reduced from O(nm) to O(n 2.82 ), and the time for finding a star cutset, a two-pair, and a dominating pair is reduced from O(nm) to O(n 2.79 ). It is also shown that each of these problems is at least as hard as one of three basic graph problems for which the best known algorithms run in times O(nm) and O(n alpha ). We note that the fast matrix multiplication algorithms do not seem to be practical because of the enormous constants needed to achieve the asymptotic time bounds. These results are important theoretically for breaking the n 3 barrier rather than giving efficient algorithms for a user.

A pair of vertices of a graph is called a dominating pair if the vertex set of every path between these two vertices is a dominating set of the graph. A graph is a weak dominating pair graph if it has a dominating pair. Further, a graph is called a dominating pair graph if each of its connected induced subgraphs is a weak dominating pair graph. Dominating pair graphs form a class of graphs containing interval, permutation, cocomparability, and asteroidal triple-free graphs. Our purpose is to study the structural properties of dominating pair graphs. Our main results are a polar theorem for the dominating pairs in weak dominating pair graphs and an existence theorem for minimum cardinality connected dominating sets that induce a simple path in connected dominating pair graphs of diameter not equal to three. Furthermore, we present a forbidden induced subgraph characterization of chordal dominating pair graphs.

We examine the bandwidth problem in circular-arc graphs, chordal graphs with a bounded number of leaves in the clique tree, and k-polygon graphs (fixed k). We show that all of these graph classes admit efficient approximation algorithms which are based on exact or approximate bandwidth layouts of related interval graphs. Specifically, we obtain a bandwidth approximation algorithm for circular-arc graphs that executes in O(n log 2 n ) time and has performance ratio 2, which is the best possible performance ratio of any polynomial time bandwidth approximation algorithm for circular-arc graphs. For chordal graphs with at most k leaves in the clique tree, we obtain a performance ratio of 2k in O(k(n+m)) time, and our algorithm for k-polygon graphs has performance ratio 2k 2 and runs in time O(n 3 ).

We consider the degree-preserving spanning tree (DPST) problem: Given a connected graph G, find a spanning tree T of G such that as many vertices of T as possible have the same degree in T as in G. This problem is a graph-theoretical translation of a problem arising in the system-theoretical context of identifiability in networks, a concept which has applications in, for example, water distribution networks and electrical networks. We show that the DPST problem is NP-complete, even when restricted to split graphs or bipartite planar graphs, but that it can be solved in polynomial time for graphs with a bounded asteroidal number and for graphs with a bounded treewidth. For the class of interval graphs, we give a linear time algorithm. For the class of cocomparability graphs, we give an O(n4) algorithm. Furthermore, we present linear time approximation algorithms for planar graphs of a worst-case performance ratio of 1 − ϵ for every ϵ > 0. © 2000 John Wiley & Sons, Inc.

An asteroidal triple (AT) is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called AT-free if it does not have an AT. We show that there is an O(n 4 ) time algorithm to compute the maximum weight of an independent set for AT-free graphs. Furthermore, we obtain O(n 4 ) time algorithms to solve the INDEPENDENT DOMINATING SET and the INDEPENDENT PERFECT DOMINATING SET problems on AT-free graphs. We also show how to adapt these algorithms such that they solve the corresponding problem for graphs with bounded asteroidal number in polynomial time. Finally, we observe that the problems CLIQUE and PARTITION INTO CLIQUES remain NP-complete when restricted to AT-free graphs.

An efficient algorithm listing all minimal vertex separators of an undirected graph is given. The algorithm needs polynomial time per separator that is found.

A vertex (edge) coloring $\phi:V\rightarrow \{1,2,\ldots ,t\}$ ($\phi':E\rightarrow \{1,2,\ldots,$ $t\}$) of a graph G=(V,E) is a vertex (edge) t-ranking if, for any two vertices (edges) of the same color, every path between them contains a vertex (edge) of larger color. The {\em vertex ranking number} $\chi_{r}(G)$ ({\em edge ranking number} $\chi_{r}'(G)$) is the smallest value of t such that G has a vertex (edge) t-ranking. In this paper we study the algorithmic complexity of the {\sc Vertex Ranking} and {\sc Edge Ranking} problems. It is shown that $\chi_{r}(G)$ can be computed in polynomial time when restricted to graphs with treewidth at most k for any fixed k. We characterize the graphs where the vertex ranking number $\chi_{r}$ and the chromatic number $\chi$ coincide on all induced subgraphs, show that $\chi_{r}(G)=\chi (G)$ implies $\chi (G)=\omega (G)$ (largest clique size), and give a formula for $\chi_{r}'(K_n)$.

In this paper, we show that the treewidth and pathwidth of a permutation graph can be computed in polynomial time. In fact we show that, for permutation graphs, the treewidth and pathwidth are equal. These results make permutation graphs one of the few nontrivial graph classes for which, at the moment, treewidth is known to be computable in polynomial time. Our algorithm, which decides whether the treewidth (pathwidth) is at most some given integer k, can be implemented to run in $O( nk )$ time when the matching diagram is given. We show that this algorithm can easily be adapted to compute the pathwidth of a permutation graph in $O( nk )$ time, where k is the pathwidth.

The authors determine the algorithmic complexity of domination and variants on cocomparability graphs, a class of perfect graphs containing both the interval and the permutation graphs. Minimum dominating, total dominating, connected dominating, and independent dominating sets can be constructed in polynomial time. On the other hand, DOMINATING CLIQUE and MINIMUM DOMINATING CLIQUE remain NP-complete on cocomparability graphs.