Reza Saei

Subset feedback vertex sets in chordal graphs

Abstract Given a graph G = ( V , E ) and a set S ⊆ V , a set U ⊆ V is a subset feedback vertex set of ( G , S ) if no cycle in G [ V ∖ U ] contains a vertex of S . The Subset Feedback Vertex Set problem takes as input G , S , and an integer k , and the question is whether ( G , S ) has a subset feedback vertex set of cardinality or weight at most k . Both the weighted and the unweighted versions of this problem are NP-complete on chordal graphs, even on their subclass split graphs. We give an algorithm with running time O ( 1.6708 n ) that enumerates all minimal subset feedback vertex sets on chordal graphs on n vertices. As a consequence, Subset Feedback Vertex Set can be solved in time O ( 1.6708 n ) on chordal graphs, both in the weighted and in the unweighted case. As a comparison, on arbitrary graphs the fastest known algorithm for these problems has O ( 1.8638 n ) running time. We also obtain that a chordal graph G has at most 1.6708 n minimal subset feedback vertex sets, regardless of S . This narrows the gap with respect to the best known lower bound of 1.5848 n on this graph class. For arbitrary graphs, the gap is substantially wider, as the best known upper and lower bounds are 1.8638 n and 1.5927 n , respectively.

Computing the metric dimension for chain graphs

The metric dimension of a graph G is the smallest size of a set R of vertices that can distinguish each vertex pair of G by the shortest-path distance to some vertex in R. Computing the metric dimension is NP-hard, even when restricting inputs to bipartite graphs. We present a linear-time algorithm for computing the metric dimension for chain graphs, which are bipartite graphs whose vertices can be ordered by neighborhood inclusion. We show how to compute the metric dimension of bipartite chain graphs.Our algorithm works in linear time even on compact representations.We also conclude combinatorial results for simple chain graphs.Our order-theoretic arguments may be useful for other problems, as well.We indicate this for some variants of metric dimension.

An exact algorithm for subset feedback vertex set on chordal graphs

Given a graph G=(V,E) and a set S⊆V, a set U⊆V is a subset feedback vertex set of (G,S) if no cycle in G[V∖U] contains a vertex of S. The Subset Feedback Vertex Set problem takes as input G, S, and an integer k, and the question is whether (G,S) has a subset feedback vertex set of cardinality or weight at most k. Both the weighted and the unweighted versions of this problem are NP-complete on chordal graphs, even on their subclass split graphs. We give an algorithm with running time O(1.6708n) that enumerates all minimal subset feedback vertex sets on chordal graphs with n vertices. As a consequence, Subset Feedback Vertex Set can be solved in time O(1.6708n) on chordal graphs, both in the weighted and in the unweighted case. On arbitrary graphs, the fastest known algorithm for the problems has O(1.8638n) running time.

Graph classes and Ramsey numbers

Abstract For a graph class G and any two positive integers i and j , the Ramsey number R G ( i , j ) is the smallest positive integer such that every graph in G on at least R G ( i , j ) vertices has a clique of size i or an independent set of size j . For the class of all graphs, Ramsey numbers are notoriously hard to determine, and they are known only for very small values of i and j . Even if we restrict G to be the class of claw-free graphs, it is highly unlikely that a formula for determining R G ( i , j ) for all values of i and j will ever be found, as there are infinitely many nontrivial Ramsey numbers for claw-free graphs that are as difficult to determine as for arbitrary graphs. Motivated by this difficulty, we establish here exact formulas for all Ramsey numbers for three important subclasses of claw-free graphs: line graphs, long circular interval graphs, and fuzzy circular interval graphs. On the way to obtaining these results, we also establish all Ramsey numbers for the class of perfect graphs. Such positive results for graph classes are rare: a formula for determining R G ( i , j ) for all values of i and j , when G is the class of planar graphs, was obtained by Steinberg and Tovey (1993), and this seems to be the only previously known result of this kind. We complement our aforementioned results by giving exact formulas for determining all Ramsey numbers for several graph classes related to perfect graphs.

Finding Disjoint Paths in Split Graphs

The well-known Disjoint Paths problem takes as input a graph G and a set of k pairs of terminals in G, and the task is to decide whether there exists a collection of k pairwise vertex-disjoint paths in G such that the vertices in each terminal pair are connected to each other by one of the paths. This problem is known to be NP-complete, even when restricted to planar graphs or interval graphs. Moreover, although the problem is fixed-parameter tractable when parameterized by k due to a celebrated result by Robertson and Seymour, it is known not to admit a polynomial kernel unless NP ⊆ coNP/poly. We prove that Disjoint Paths remains NP-complete on split graphs, and show that the problem admits a kernel with O(k2) vertices when restricted to this graph class. We furthermore prove that, on split graphs, the edge-disjoint variant of the problem is also NP-complete and admits a kernel with O(k3) vertices. To the best of our knowledge, our kernelization results are the first non-trivial kernelization results for these problems on graph classes.

Maximal Induced Matchings in Triangle‐Free Graphs

An induced matching in a graph is a set of edges whose endpoints induce a \(1\)-regular subgraph. It is known that every \(n\)-vertex graph has at most \(10^{n/5}\approx 1.5849^n\) maximal induced matchings, and this bound is best possible. We prove that every \(n\)-vertex triangle-free graph has at most \(3^{n/3}\approx 1.4423^n\) maximal induced matchings, and this bound is attained by every disjoint union of copies of the complete bipartite graph \(K_{3,3}\). Our result implies that all maximal induced matchings in an \(n\)-vertex triangle-free graph can be listed in time \(O(1.4423^n)\), yielding the fastest known algorithm for finding a maximum induced matching in a triangle-free graph.

Ramsey Numbers for Line Graphs and Perfect Graphs

For any graph class \({\cal G}\) and any two positive integers i and j, the Ramsey number \(R_{\cal G}(i,j)\) is the smallest integer such that every graph in \({\cal G}\) on at least \(R_{\cal G}(i,j)\) vertices has a clique of size i or an independent set of size j. For the class of all graphs Ramsey numbers are notoriously hard to determine, and the exact values are known only for very small integers i and j. For planar graphs all Ramsey numbers can be determined by an exact formula, whereas for claw-free graphs there exist Ramsey numbers that are as difficult to determine as for arbitrary graphs. No further graph classes seem to have been studied for this purpose. Here, we give exact formulas for determining all Ramsey numbers for various classes of graphs. Our main result is an exact formula for all Ramsey numbers for line graphs, which form a large subclass of claw-free graphs and are not perfect. We obtain this by proving a general result of independent interest: an upper bound on the number of edges any graph can have if it has bounded degree and bounded matching size. As complementary results, we determine all Ramsey numbers for perfect graphs and for several subclasses of perfect graphs.