Polynomial fixed-parameter algorithms : a case study for longest path on interval graphs.

We study the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time. The main motivation is to get more efficient algorithms for problems with unattractive polynomial running times. Here, we focus on a fundamental graph problem: Longest Path; it is NP-hard in general but known to be solvable in O(n^4) time on n-vertex interval graphs. We show how to solve Longest Path on Interval Graphs, parameterized by vertex deletion number k to proper interval graphs, in O(k^9n) time. Notably, Longest Path is trivially solvable in linear time on proper interval graphs, and the parameter value k can be approximated up to a factor of 4 in linear time. From a more general perspective, we believe that using parameterized complexity analysis for polynomial-time solvable problems offers a very fertile ground for future studies for all sorts of algorithmic problems. It may enable a refined understanding of efficiency aspects for polynomial-time solvable problems, similarly to what classical parameterized complexity analysis does for NP-hard problems

New geometric representations and domination problems on tolerance and multitolerance graphs

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain amount of overlap without being in conflict. In one of the most natural generalizations of tolerance graphs with direct applications in the comparison of DNA sequences from different organisms, namely multitolerance graphs, two tolerances are allowed for each interval – one from the left and one from the right side. Several efficient algorithms for optimization problems that are NPhard in general graphs have been designed for tolerance and multitolerance graphs. In spite of this progress, the complexity status of some fundamental algorithmic problems on tolerance and multitolerance graphs, such as the dominating set problem, remained unresolved until now, three decades after the introduction of tolerance graphs. In this article we introduce two new geometric representations for tolerance and multitolerance graphs, given by points and line segments in the plane. Apart from being important on their own, these new representations prove to be a powerful tool for deriving both hardness results and polynomial time algorithms. Using them, we surprisingly prove that the dominating set problem can be solved in polynomial time on tolerance graphs and that it is APX-hard on multitolerance graphs, solving thus a longstanding open problem. This problem is the first one that has been discovered with a different complexity status in these two graph classes. Furthermore we present an algorithm that solves the independent dominating set problem on multitolerance graphs in polynomial time, thus demonstrating the potential of this new representation for further exploitation via sweep line algorithms

Polynomial fixed-parameter algorithms: A case study for longest path on interval graphs

We study the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time. The main motivation is to get more efficient algorithms for problems with unattractive polynomial running times. Here, we focus on a fundamental graph problem: Longest Path; it is NP-hard in general but known to be solvable in O(n^4) time on n-vertex interval graphs. We show how to solve Longest Path on Interval Graphs, parameterized by vertex deletion number k to proper interval graphs, in O(k^9n) time. Notably, Longest Path is trivially solvable in linear time on proper interval graphs, and the parameter value k can be approximated up to a factor of 4 in linear time. From a more general perspective, we believe that using parameterized complexity analysis for polynomial-time solvable problems offers a very fertile ground for future studies for all sorts of algorithmic problems. It may enable a refined understanding of efficiency aspects for polynomial-time solvable problems, similarly to what classical parameterized complexity analysis does for NP-hard problems

Graph Minors and Parameterized Algorithm Design

Abstract. The Graph Minors Theory, developed by Robertson and Sey-mour, has been one of the most influential mathematical theories in pa-rameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct meta-algorithmic consequences, we present the algorithmic appli-cations of core theorems such as the grid-exclusion theorem, and we give a brief description of the irrelevant vertex technique

Obstructions for Tree-depth

For every k ≥ 0, we define Gk as the class of graphs with tree-depth at most k, i.e. the class containing every graph G admitting a valid colouring ρ : V (G) → {1, ..., k} such that every (x, y)-path between two vertices where ρ (x) = ρ (y) contains a vertex z where ρ (z) > ρ (x). In this paper we study the class obs (Gk) of minor-minimal elements not belonging in Gk for every k ≥ 0. We give a precise characterization of Gk, k ≤ 3 and prove a structural lemma for creating graphs G ∈ obs (Gk), k > 0. As a consequence, we obtain a precise characterization of all acyclic graphs in obs (Gk) and we prove that they are exactly frac(1, 2) 22k - 1 - k (1 + 22k - 1 - k). © 2009 Elsevier B.V. All rights reserved

A min-max theorem for LIFO-search

We introduce a variant of the classic node search game called LIFO-search where searchers are assigned different numbers. The additional rule is that a searcher can be removed only if no searchers of lower rank are in the graph at that moment. We introduce the notion of shelters in graphs and we prove a min-max theorem implying their equivalence with the tree-depth parameter. As shelters provide escape strategies for the fugitive, this implies that the LIFO-search game is monotone and that the LIFO-search parameter is equivalent with the one of tree-depth. © 2011 Elsevier B.V

The Flat Wall Theorem for Bipartite Graphs with Perfect Matchings

Matching minors are a specialised version of minors fit for the study of graphs with perfect matchings. The first major appearance of matching minors was in a result by Little who showed that a bipartite graph is Pfaffian if and only if it does not contain K3, 3 as a matching minor. Later it was shown, that K3, 3 -matching minor free bipartite graphs are essentially, that is after some clean-up and with a single exception, bipartite planar graphs glued together at 4-cycles. We generalise these ideas by giving an approximate description of bipartite graphs excluding Kt , t as a matching minor in the spirit of the famous Flat Wall Theorem of Robertson and Seymour. In essence, we show that every bipartite Kt , t -matching minor free graph is locally K3, 3 -matching minor free after removing an apex set of bounded size. © 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG

Forbidding Kuratowski graphs as immersions

Immersion is a containment relation on graphs that is weaker than topological minor. (Every topological minor of a graph is also its immersion.) The graphs that do not contain any of the Kuratowski graphs (K5 and K3,3) as topological minors are exactly planar graphs. We give a structural characterization of graphs that exclude the Kuratowski graphs as immersions. We prove that they can be constructed from planar graphs that are subcubic or of branch-width at most 10 by repetitively applying i-edge-sums, for i ∈ {1, 2, 3}. We also use this result to give a structural characterization of graphs that exclude K3,3 as an immersion. © 2014 Wiley Periodicals, Inc

Excluding graphs as immersions in surface embedded graphs

We prove a structural characterization of graphs that forbid a fixed graph H as an immersion and can be embedded in a surface of Euler genus γ. In particular, we prove that a graph G that excludes some connected graph H as an immersion and is embedded in a surface of Euler genus γ has either "small" treewidth (bounded by a function of H and γ) or "small" edge connectivity (bounded by the maximum degree of H). Using the same techniques we also prove an excluded grid theorem on bounded genus graphs for the immersion relation. © 2013 Springer-Verlag

Effective computation of immersion obstructions for unions of graph classes

In the final paper of the Graph Minors series [Neil Robertson and Paul D. Seymour. Graph minors XXIII. Nash-Williams' immersion conjecture J. Comb. Theory, Ser. B, 100(2):181-205, 2010.], N. Robertson and P. Seymour proved that graphs are well-quasi-ordered with respect to the immersion relation. A direct implication of this theorem is that each class of graphs that is closed under taking immersions can be fully characterized by forbidding a finite set of graphs (immersion obstruction set). However, as the proof of the well-quasi-ordering theorem is non-constructive, there is no generic procedure for computing such a set. Moreover, it remains an open issue to identify for which immersion-closed graph classes the computation of those sets can become effective. By adapting the tools that where introduced in [Isolde Adler, Martin Grohe and Stephan Kreutzer. Computing excluded minors, SODA, 2008: 641-650.] for the effective computation of obstruction sets for the minor relation, we expand the horizon of the computability of obstruction sets for immersion-closed graph classes. In particular, we prove that there exists an algorithm that, given the immersion obstruction sets of two graph classes that are closed under taking immersions, outputs the immersion obstruction set of their union. © 2012 Springer-Verlag