Linear Recognition of Almost Interval Graphs

Abstract

Let interval + kv, interval + ke, and interval − ke denote the classes of graphs that can be obtained from some interval graph by adding k vertices, adding k edges, and deleting k edges, respectively. When k is small, these graph classes are called almost interval graphs. They are well motivated from computational biology, where the data ought to be represented by an interval graph while we can only expect an almost interval graph for the best. For any fixed k, we give linear-time algorithms for recognizing all these classes, and in the case of membership, our algorithms provide also a specific interval graph as evidence. When k is part of the input, these problems are also known as graph modification problems, all NP-complete. Our results imply that they are fixed-parameter tractable parameterized by k, thereby resolving the long-standing open problem on the parameterized complexity of recognizing interval + ke, first asked by Bodlaender et al. [Bioinformatics, 11:49-57, 1995]. Moreover, our algorithms for recognizing interval + kv and interval − ke run in times O( 6k • (n + m)) and O(8 k • (n + m)), (where n and m stand for the numbers of vertices and edges respectively in the input graph,) significantly improving the O(k 2k • n 3 m)-time algorithm of Heggernes et al. [STOC 2007] and the O(10 k • n 9 )-time algorithm of Cao and Marx [SODA 2014] respectively. * Dedicated to Jianer Chen on the occasion of his 60th birthday.

Chordal Editing is Fixed-Parameter Tractable

“…The fixed-parameter tractability of their completion versions were shown by Kaplan et al [13] and Villanger et al [26]; their vertex deletion versions were shown by van 't Hof and Villanger [25] and Cao and Marx [7]. A very recent result of Cao [5] complemented them by showing that the edge deletion versions are FPT as well.…”

Forbidden induced subgraphs of normal Helly circular-arc graphs: Characterization and detection

“…Recall that Lin et al [10] have given a linear-time algorithm for verifying whether a circular-arc model is normal and Helly. In fact, given a circular-arc model, we can in linear time find the minimum number of arcs that cover the circle [2]. Proof of Theorem 1.3.…”

“…The crucial idea behind our certifying algorithm is a novel correlation between normal Helly circular-arc graphs and interval graphs, which can be efficiently used for algorithmic purpose. This was originally proposed in the detection of small forbidden induced subgraph of interval graphs [2], i.e., the opposite direction of the current paper. In particular, in [2] we have used a similar definition of the auxiliary graph and pertinent observations.…”

“…This was originally proposed in the detection of small forbidden induced subgraph of interval graphs [2], i.e., the opposite direction of the current paper. In particular, in [2] we have used a similar definition of the auxiliary graph and pertinent observations. However, the main structures and the procedures for the detection of forbidden induced subgraphs divert completely.…”

“…However, the main structures and the procedures for the detection of forbidden induced subgraphs divert completely. For example, the most common forbidden induced subgraphs in [2] are 4-and 5-holes, which, however, are allowed in normal Helly circular-arc graphs. This means that the interaction between N[v] and G − N[v] are far more subtle, and thus the detection of minimal forbidden induced subgraphs in the current paper is significantly more complicated than that of [2].…”