Efficient algorithms for domination and Hamilton circuit problems on permutation graphs

Domination in permutation graphs

Mind the independence gap

On polygon numbers of circle graphs and distance hereditary graphs

SummaryThe domination problem is one of the fundamental graph problems, and there are many variations. In this article, we propose a new problem called the minus ‐domination problem where , and are integers such that , , and . The problem is to assign a value from for each vertex in a graph such that the local summation of values is greater than or equal to . We also propose a framework named the bounded lattice domination for a class of domination problems, including the minus ‐domination problem. Then, we present a self‐stabilizing distributed algorithm under the distance‐2 model for the bounded lattice domination. Here, self‐stabilization is a class of fault‐tolerant distributed algorithms that tolerate transient faults. The time complexity for convergence is , where is the number of processes in a network if the cardinality of the domain of process values is finite and constant. Otherwise, the time complexity for convergence is .

For a simple, undirected, connected graph G, a function h : V → {0, 1, 2} is called a total Roman {2}-dominating function (TR2DF) if for every vertex v in V with weight 0, either there exists a vertex u in NG(v) with weight 2, or at least two vertices x, y in NG(v) each with weight 1, and the subgraph induced by the vertices with weight more than zero has no isolated vertices. The weight of TR2DF h is ∑peV h(p). The problem of determining TR2DF of minimum weight is called minimum total Roman {2}-domination problem (MTR2DP). We show that MTR2DP is polynomial time solvable for bounded treewidth graphs, threshold graphs and chain graphs. We design a 2(ln(Δ - 0.5) + 1.5)-approximation algorithm for the MTR2DP and show that the same cannot have (1-δ) ln(|V|) ratio approximation algorithm for any δ > 0 unless P=Np . Next, we show that MTR2DP is APX-hard for graphs with Δ=4. We also show that the domination and TR2DF problems are not equivalent in computational complexity aspects.

Consider a class of graphs $$\mathcal{G}$$ having a polynomial time algorithm computing the set of all minimal separators for every graph in $$\mathcal{G}$$ . We show that there is a polynomial time algorithm for treewidth and minimum fill-in, respectively, when restricted to the class $$\mathcal{G}$$ . Many interesting classes of intersection graphs have a polynomial time algorithm computing all minimal separators, like permutation graphs, circle graphs, circular arc graphs, distance hereditary graphs, chordal bipartite graphs etc. Our result generalizes earlier results for the treewidth and minimum fill-in for several of these classes. We also consider the related problems pathwidth and interval completion when restricted to some special graph classes.

FPT algorithms for domination in sparse graphs and beyond

On the complexity of the black-and-white coloring problem on some classes of perfect graphs

We reduce (in polynomial time) the enumeration of minimal dominating sets in interval and permutation graphs to the enumeration of paths in directed acyclic graphs(DAGs). As a consequence, we can enumerate with linear delay, after a polynomial time pre-processing, minimal dominating sets in interval and permutation graphs. We can also count them in polynomial time. This improves considerably upon previously known results on interval graphs, and up to our knowledge no output polynomial time algorithm for the enumeration of minimal dominating sets and their counting were known for permutation graphs.

Finding dominating induced matchings in [formula omitted]-free graphs in polynomial time

The last chapter of this book eventually presents some famous classes of perfect graphs, namely comparability, interval, permutation, and split graphs. We first present an example of a graph which is at the same time a triangulated, a comparability, a split, and a permutation graph. The importance to be an interval is illustrated with Berge’s mystery story. We discuss furthermore the generation of permutation graphs and close with how to recognise that a given graph is in fact a permutation graph.

Exploring the complexity boundary between coloring and list-coloring

A subset T⊆V(G) of vertices of a graph G is said to be cyclable if G has a cycle C containing every vertex of T, and for a positive integer k, a graph G is k-cyclable if every set of vertices of size at most k is cyclable. The Terminal Cyclability problem asks, given a graph G and a set T of vertices, whether T is cyclable, and the k-Cyclability problem asks, given a graph G and a positive integer k, whether G is k-cyclable. These problems are generalizations of the classical Hamiltonian Cycle problem. We initiate the study of these problems for graph classes that admit polynomial algorithms for Hamiltonian Cycle. We show that Terminal Cyclability can be solved in linear time for interval graphs, bipartite permutation graphs and cographs. Moreover, we construct certifying algorithms that either produce a solution, that is a cycle, or output a graph separator that certifies a no-answer. We use these results to show that k-Cyclability can be solved in polynomial time when restricted to the aforementioned graph classes.

For a simple, undirected, connected graph [Formula: see text], a function [Formula: see text] which satisfies the following conditions is called a total Roman {3}-dominating function (TR3DF) of [Formula: see text] with weight [Formula: see text]: (C1) For every vertex [Formula: see text] if [Formula: see text], then [Formula: see text] has [Formula: see text] ([Formula: see text]) neighbors such that whose sum is at least 3, and if [Formula: see text], then [Formula: see text] has [Formula: see text] ([Formula: see text]) neighbors such that whose sum is at least 2. (C2) The subgraph induced by the set of vertices labeled one, two or three has no isolated vertices. For a graph [Formula: see text], the smallest possible weight of a TR3DF of [Formula: see text] denoted [Formula: see text] is known as the total Roman[Formula: see text]-domination number of [Formula: see text]. The problem of determining [Formula: see text] of a graph [Formula: see text] is called minimum total Roman {3}-domination problem (MTR3DP). In this paper, we show that the problem of deciding if [Formula: see text] has a TR3DF of weight at most [Formula: see text] for chordal graphs is NP-complete. We also show that MTR3DP is polynomial time solvable for bounded treewidth graphs, chain graphs and threshold graphs. We design a [Formula: see text]-approximation algorithm for the MTR3DP and show that the same cannot have [Formula: see text] ratio approximation algorithm for any [Formula: see text] unless NP [Formula: see text]. Next, we show that MTR3DP is APX-complete for graphs with [Formula: see text]. We also show that the domination and total Roman {3}-domination problems are not equivalent in computational complexity aspects. Finally, we present an integer linear programming formulation for MTR3DP.

A set of vertices in a graph is c-colorable if the subgraph induced by the set has a proper c-coloring. In this paper, we study the problem of finding a step-by-step transformation (called a reconfiguration sequence) between two c-colorable sets in the same graph. This problem generalizes the well-studied Independent Set Reconfiguration problem. As the first step toward a systematic understanding of the complexity of this general problem, we study the problem on classes of perfect graphs. We first focus on interval graphs and give a combinatorial characterization of the distance between two c-colorable sets. This gives a linear-time algorithm for finding an actual shortest reconfiguration sequence for interval graphs. Since interval graphs are exactly the graphs that are simultaneously chordal and co-comparability, we then complement the positive result by showing that even deciding reachability is PSPACE-complete for chordal graphs and for co-comparability graphs. The hardness for chordal graphs holds even for split graphs. We also consider the case where c is a fixed constant and show that in such a case the reachability problem is polynomial-time solvable for split graphs but still PSPACE-complete for co-comparability graphs. The complexity of this case for chordal graphs remains unsettled. As by-products, our positive results give the first polynomial-time solvable cases (split graphs and interval graphs) for Feedback Vertex Set Reconfiguration.