Structural Parameterizations with Modulator Oblivion

It is known that problems like Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal are polynomial time solvable in the class of chordal graphs. We consider these problems in a graph that has at most k vertices whose deletion results in a chordal graph, when parameterized by k. While this investigation fits naturally into the recent trend of what are called "structural parameterizations", here we assume that the deletion set is not given. One method to solve them is to compute a k-sized or an approximate (f(k) sized, for a function f) chordal vertex deletion set and then use the structural properties of the graph to design an algorithm. This method leads to at least k^O(k)n^O(1) running time when we use the known parameterized or approximation algorithms for finding a k-sized chordal deletion set on an n vertex graph. In this work, we design 2^O(k)n^O(1) time algorithms for these problems. Our algorithms do not compute a chordal vertex deletion set (or even an approximate solution). Instead, we construct a tree decomposition of the given graph in time 2^O(k)n^O(1) where each bag is a union of four cliques and O(k) vertices. We then apply standard dynamic programming algorithms over this special tree decomposition. This special tree decomposition can be of independent interest. Our algorithms are, what are sometimes called permissive in the sense that given an integer k, they detect whether the graph has no chordal vertex deletion set of size at most k or output the special tree decomposition and solve the problem. We also show lower bounds for the problems we deal with under the Strong Exponential Time Hypothesis (SETH)

Role coloring bipartite graphs

A k-role coloring of a graph G is an assignment of colors to the vertices of G such that every color is used at least once and if any two vertices are assigned the same color, then their neighborhood are assigned the same set of colors. By definition, every graph on n vertices admits an n-role coloring. While for every graph on n vertices, it is trivial to decide if it admits a 1-role coloring, determining whether a graph admits a k-role coloring is a notoriously hard problem for k greater than or equal to 2. In fact, it is known that k-Role coloring is NP-complete for k at least 2 on general graph class. There has been extensive research on the complexity of k-role coloring on various hereditary graph classes. Furthering this direction of research, we show that k-Role coloring is NP-complete on bipartite graphs for k at least 3 (while it is trivial for k=2). We complement the hardness result by characterizing 3-role colorable bipartite chain graphs, leading to a polynomial time algorithm for 3-Role coloring for this class of graphs. We further show that 2-Role coloring is NP-complete for graphs that are d vertices or edges away from the class of bipartite graphs, even when d=1

Structural Parameterizations with Modulator Oblivion

It is known that problems like Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal are polynomial time solvable in the class of chordal graphs. We consider these problems in a graph that has at most k vertices whose deletion results in a chordal graph when parameterized by k. While this investigation fits naturally into the recent trend of what is called ‘structural parameterizations’, here we assume that the deletion set is not given. One method to solve them is to compute a k-sized or an approximate (f(k) sized, for a function f) chordal vertex deletion set and then use the structural properties of the graph to design an algorithm. This method leads to at least kO(k)nO(1) running time when we use the known parameterized or approximation algorithms for finding a k-sized chordal deletion set on an n vertex graph. In this work, we design 2 O(k)nO(1) time algorithms for these problems. Our algorithms do not compute a chordal vertex deletion set (or even an approximate solution). Instead, we construct a tree decomposition of the given graph in 2 O(k)nO(1) time where each bag is a union of four cliques and O(k) vertices. We then apply standard dynamic programming algorithms over this special tree decomposition. This special tree decomposition can be of independent interest. Our algorithms are, what are sometimes called permissive in the sense that given an integer k, they detect whether the graph has no chordal vertex deletion set of size at most k or output the special tree decomposition and solve the problem. We also show lower bounds for the problems we deal with under the strong exponential time hypothesis. © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature

Role coloring bipartite graphs

A k-role coloring of a graph G is an assignment of colors to the vertices of G such that every color is used at least once and if any two vertices are assigned the same color, then their neighborhood are assigned the same set of colors. By definition, every graph on n vertices admits an n-role coloring. While for every graph on n vertices, it is trivial to decide if it admits a 1-role coloring, determining whether a graph admits a k-role coloring is a notoriously hard problem for k greater than or equal to 2. In fact, it is known that k-Role coloring is NP-complete for k at least 2 on general graph class. There has been extensive research on the complexity of k-role coloring on various hereditary graph classes. Furthering this direction of research, we show that k-Role coloring is NP-complete on bipartite graphs for k at least 3 (while it is trivial for k=2). We complement the hardness result by characterizing 3-role colorable bipartite chain graphs, leading to a polynomial time algorithm for 3-Role coloring for this class of graphs. We further show that 2-Role coloring is NP-complete for graphs that are d vertices or edges away from the class of bipartite graphs, even when d=1