On-line interval graphs coloring — Modification of the First-Fit algorithm and its performance ratio

We propose a modification of the First-Fit algorithm in which we forbid the use of the most frequently used color. We provide some lower bound of the performance ratio [Formula: see text] on the family of interval graphs [Formula: see text], where [Formula: see text] denotes the number of used colors by the modified version of the First-Fit algorithm and [Formula: see text] is the number of vertices in the largest complete subgraph in [Formula: see text]. We also compare the modification with the usual First-Fit algorithm on some experimental data.

$${\mathcal {U}}$$-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited

Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs—a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a richer class of graphs. In particular, mixed unit interval graphs may contain a claw as an induced subgraph, as opposed to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyacı et al. (Inf Process Lett 121:29–33, 2017. 10.1016/j.ipl.2017.01.007). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs.

On the $e$-Positivity of $(claw, 2K_2)$-Free Graphs

Motivated by Stanley and Stembridge's conjecture about the $e$-positivity of claw-free incomparability graphs, Hamel and her collaborators studied the $e$-positivity of $(claw, H)$-free graphs, where $H$ is a four-vertex graph. In this paper we establish the $e$-positivity of generalized pyramid graphs and $2K_2$-free unit interval graphs, which are two important families of $(claw, 2K_2)$-free graphs. Hence we affirmatively solve one problem proposed by Hamel, Hoáng and Tuero, and another problem considered by Foley, Hoáng and Merkel.

The Steiner cycle and path cover problem on interval graphs

The Steiner path problem is a common generalization of the Steiner tree and the Hamiltonian path problem, in which we have to decide if for a given graph there exists a path visiting a fixed set of terminals. In the Steiner cycle problem we look for a cycle visiting all terminals instead of a path. The Steiner path cover problem is an optimization variant of the Steiner path problem generalizing the path cover problem, in which one has to cover all terminals with a minimum number of paths. We study those problems for the special class of interval graphs. We present linear time algorithms for both the Steiner path cover problem and the Steiner cycle problem on interval graphs given as endpoint sorted lists. The main contribution is a lemma showing that backward steps to non-Steiner intervals are never necessary. Furthermore, we show how to integrate this modification to the deferred-query technique of Chang et al. to obtain the linear running times.

Efficient enumeration of non-isomorphic interval graphs

Recently, Yamazaki et al. provided an algorithm that enumerates all non-isomorphic interval graphs on $n$ vertices with an $O(n^4)$ time delay. In this paper, we improve their algorithm and achieve $O(n^3 \log n)$ time delay. We also extend the catalog of these graphs providing a list of all non-isomorphic interval graphs for all $n$ up to $15$.