Total k-Domatic Partition on Some Classes of Graphs

A bipartite graph G(X, Y) whose vertex set is partitioned into X and Y is a convex bipartite graph, if there is an ordering of $$X=(x_1,\ldots ,x_m)$$ such that for all $$y \in Y$$ , $$N_G(y)$$ is consecutive with respect to the ordering of X, and G is said to have convexity with respect to X. A k-star caterpillar is a tree with a collection of stars with each star having k vertices of degree one whose roots are joined by a path. For a bipartite graph with partitions X and Y, we associate a k-star caterpillar on X such that for each vertex in Y, its neighborhood induces a tree. The minimum Steiner tree problem (STREE) is defined as follows: given a connected graph $$G=(V,E)$$ and a subset of vertices $$R \subseteq V(G)$$ , the objective is to find a minimum cardinality set $$S \subseteq V(G)$$ such that the set $$R \cup S$$ induces a connected subgraph. In this paper, we present the following dichotomy result: we show that STREE is NP-complete for 1-star caterpillar convex bipartite graphs and polynomial-time solvable for 0-star caterpillar convex bipartite graphs (also known as convex bipartite graphs). We also strengthen the well-known result of Müller and Brandstädt (Theoret Comput Sci 53(2-3):257-265, 1987), which says STREE in chordal bipartite graphs is NP-complete (reduction instances are 3-star caterpillar convex bipartite graphs). As an application, we use our STREE results to solve: (i) the classical dominating set problem in convex bipartite graphs, (ii) STREE on interval graphs, linear time.

For any class C of bipartite graphs, we define quasi-C to be the class of all graphs G such that every bipartition of G belongs to C. This definition is motivated by a generalisation of the switch Markov chain on perfect matchings from bipartite graphs to nonbipartite graphs. The monotone graphs, also known as bipartite permutation graphs and proper interval bigraphs, are such a class of bipartite graphs. We investigate the structure of quasi-monotone graphs and hence construct a polynomial time recognition algorithm for graphs in this class.

Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances

We resolve a number of long-standing open problems in online graph coloring. More specifically, we develop tight lower bounds on the performance of online algorithms for fundamental graph classes. An important contribution is that our bounds also hold for randomized online algorithms, for which hardly any results were known. Technically, we construct lower bounds for chordal graphs. The constructions then allow us to derive results on the performance of randomized online algorithms for the following further graph classes: trees, planar, bipartite, inductive, bounded-treewidth and disk graphs. It shows that the best competitive ratio of both deterministic and randomized online algorithms is Theta(log n), where n is the number of vertices of a graph. Furthermore, we prove that this guarantee cannot be improved if an online algorithm has a lookahead of size O(n/log n) or access to a reordering buffer of size n^(1-epsilon), for any 0 < epsilon <= 1. A consequence of our results is that, for all of the above mentioned graph classes except bipartite graphs, the natural First Fit coloring algorithm achieves an optimal performance, up to constant factors, among deterministic and randomized online algorithms.

An independent dominating set in a graph is a subset of vertices, such that every vertex outside this subset has a neighbor in this subset (dominating), and the induced subgraph of this subset contains no edge (independent). It was known that finding the minimum independent dominating set (Independent Domination) is \(\cal{NP}\)-complete on bipartite graphs, but tractable on convex bipartite graphs. A bipartite graph is called tree convex, if there is a tree defined on one part of the vertices, such that for every vertex in another part, the neighborhood of this vertex is a connected subtree. A convex bipartite graph is just a tree convex one where the tree is a path. We find that the sum of larger-than-two degrees of the tree is a key quantity to classify the computational complexity of independent domination on tree convex bipartite graphs. That is, when the sum is bounded by a constant, the problem is tractable, but when the sum is unbounded, and even when the maximum degree of the tree is bounded, the problem is \(\cal{NP}\)-complete.KeywordsBipartite GraphMaximum DegreeChordal GraphMinimum CardinalitySatisfying AssignmentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A proper k-coloring of a graph is an assignment of colors from the set $$\{1,2,\ldots ,k\}$$ to the vertices of the graph such that no two adjacent vertices receive the same color. Given a graph $$G=(V,E)$$ , the Dynamic Coloring problem asks to find a proper k-coloring of G such that for every vertex $$v\in V(G)$$ of degree at least two, there exists at least two distinct colors appearing in the neighborhood of v. The minimum integer k such that there is a dynamic coloring of G using k colors is called the dynamic chromatic number of G and is denoted by $$\chi _d(G)$$ . The problem is NP-complete in general, but solvable in polynomial time on several restricted families of graphs. In this paper, we study the problem on restricted classes of graphs. We show that the problem can be solved in polynomial time on chordal graphs and biconvex bipartite graphs. On the other hand, we show that it is NP-complete on star-convex bipartite graphs, comb-convex bipartite graphs and perfect elimination bipartite graphs. Next, we initiate the study on Dynamic Coloring from the parameterized complexity perspective. First, we show that the problem is fixed-parameter tractable when parameterized by neighborhood diversity or twin-cover. Then, we show that the problem is fixed-parameter tractable when parameterized by the combined parameters clique-width and the number of colors.

Classes of bipartite graphs related to chordal graphs

A graph G is said to be a bisplit graph if its vertex set can be partitioned into a stable set and a complete bipartite graph. The minimum Steiner tree problem (STREE) is defined as follows: given a connected graph G and a subset of vertices \(R\subseteq V(G)\), the objective is to find a minimum cardinality set \(S\subset V (G)\) such that the set \(R \cup S\) induces a connected subgraph. In this paper, we present an interesting dichotomy result for STREE on bisplit graphs, we show that STREE is polynomial-time solvable for chordal bipartite bisplit graphs, and NP-complete otherwise. Further, we show that for chordal bisplit graphs, the problem is polynomial-time solvable. A revisit of our NP-complete reduction instances reveals that the instances are diameter at most 5 bipartite graphs. We also obtain one more dichotomy result for STREE on bisplit graphs which says that for diameter 5 the problem is NP-complete and polynomial-time solvable for diameter at most 4. On the parameterized complexity front, we show that the parameterized version of Steiner tree problem on bisplit graphs is fixed-parameter tractable when the parameter is the biclique size and is W[2]-hard on bisplit graphs if the parameter is the solution size. We conclude this paper by presenting structural results of bisplit graphs which will be of use to solve other combinatorial problems.KeywordsBisplit graphsThe Steiner tree problemW-hardnessDichotomy results

We strengthen a result by Laskar and Lyle (Discrete Appl. Math. (2009), 330–338) by proving that it is \(\textsc {NP}\)-complete to decide whether a bipartite planar graph can be partitioned into three independent dominating sets. In contrast, we show that this is always possible for every maximal outerplanar graph with at least three vertices. Moreover, we extend their previous result by proving that deciding whether a bipartite graph can be partitioned into k independent dominating sets is \(\textsc {NP}\)-complete for every \(k \ge 3\). We also strengthen a result by Henning et al. (Discrete Math. (2009), 6451–6458) by showing that it is \(\textsc {NP}\)-complete to determine if a graph has two disjoint independent dominating sets, even when the problem is restricted to triangle-free planar graphs. Finally, for every \(k \ge 3\), we show that there is some constant t depending only on k such that deciding whether a k-regular graph can be partitioned into t independent dominating sets is \(\textsc {NP}\)-complete. We conclude by deriving moderately exponential-time algorithms for the problem.

Graph isomorphism completeness for chordal bipartite graphs and strongly chordal graphs

Random generation and enumeration of bipartite permutation graphs

The boxicity of a graph G is defined as the minimum integer k such that G is an intersection graph of axis-parallel k-dimensional boxes. Chordal bipartite graphs are bipartite graphs that do not contain an induced cycle of length greater than 4. It was conjectured by Otachi, Okamoto and Yamazaki that chordal bipartite graphs have boxicity at most 2. We disprove this conjecture by exhibiting an infinite family of chordal bipartite graphs that have unbounded boxicity.

Parity graphs form a superclass of bipartite graphs. Since their introduction, all the algorithms proposed as solutions to the recognition problem and other computational problems exploit the structural property described by Burlet and Uhry in [4].This paper newly describes a different structural property based on split decomposition for parity graphs. This result, together with the observation that the split decomposition process can be performed in linear time, allows us to provide optimum algorithms for both the recognition problem and the maximum weighted clique problem in this class. We further propose a general algorithm to solve the maximum weighted independent set problem for certain classes of graphs. Its application to parity graphs provides the best solution for this problem. Moreover, when the algorithm is applied to distance-hereditary graphs, it equals the best known solution, which is also the optimum for this class.A remarkable consequence of this work is that the extension of bipartite graphs to parity graphs does not increase the complexity of these basic problems, since the worst case occurs when the parity graph is an undecomposable bipartite graph.

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.

Motivated by the concept of reverse signed domination, we introduce the reverse minus domination problem on graphs, and study the reverse minus and signed domination problems from the algorithmic point of view. In this paper, we show that both the reverse minus and signed domination problems are polynomial-time solvable for strongly chordal graphs and distance-hereditary graphs, and are linear-time solvable for trees. For chordal graphs and bipartite planar graphs, however, we show that the decision problem corresponding to the reverse minus domination problem is NP-complete. For doubly chordal graphs and bipartite planar graphs, we show that the decision problem corresponding to the reverse signed domination problem is NP-complete. Furthermore, we show that even when restricted to bipartite planar graphs or doubly chordal graphs, the reverse signed domination problem is not fixed parameter tractable.