The Independent Set problem is NP-hard in general, however polynomial time algorithms exist for the problem on various specific graph classes. Over the last couple of decades there has been a long sequence of papers exploring the boundary between the NPhard and polynomial time solvable cases. In particular the complexity of Independent Set on P5-free graphs has received significant attention, a...

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, that is, given an undirected graph, find a maximum-length path in G. Longest Path is NP-hard in general but kno...

In social networks the Strong Triadic Closure is an assignment of the edges with strong or weak labels such that any two vertices that have a common neighbor with a strong edge are adjacent. The problem of maximizing the number of strong edges that satisfy the strong triadic closure was recently shown to be NP-complete for general graphs. Here we initiate the study of graph classes for which th...

We show that a quotient group of a CI-group with respect to (di)graphs is a CI-group with respect to (di)graphs. In [1,2], Babai and Frankl provided strong constraints on which finite groups could be CI-groups with respect to graphs. As a tool in this program, they proved [1, Lemma 3.5] that a quotient group G/N of a CI-group G with respect to graphs is a CI-group with respect to graphs provide...

The vertex colouring problem is known to be NP-complete in the class of triangle-free graphs. Moreover, it remains NP-complete even if we additionally exclude a graph F which is not a forest. We study the computational complexity of the problem in (K3, F )-free graphs with F being a forest. From known results it follows that for any forest F on 5 vertices, the vertex colouring problem is polyno...

A connection tree of a graph G for terminal set W is subgraph T such that leaves( ) ⊆ V(T) . non-terminal vertex called linker if its degree in exactly 2, and it router at least 3. The erminal problem (TCP) asks whether admits with most ℓ linkers r routers, while the S teiner k vertices. We prove that, ≥ 1 fixed, then TCP polynomial-time solvable when restricted to split graphs. This result sep...

HT-graphs have been introduced in [11] and investigated with respect to location problems on graphs. In this paper two new characterizations of these graphs are given and then it is shown that the central vertex, connected r-domination and Steiner trees problems are linear or almost linear time solvable in HT-graphs.

The class of problems solvable in logarithmic space has recently replenished with the isomorphism testing for interval graphs. We discuss this result, prospects of its extension to larger classes of graphs, and related issues such as constructing canonical models of intersection graphs and solving the Star System Problem for restricted classes of graphs.

We consider the problem of partitioning bounded degree graphs into triangles. We show that this problem is polynomial time solvable on graphs of maximum degree three by giving a linear time algorithm. We also show that this problem becomes NP-complete on graphs of maximum degree four. Moreover, we show that there is no subexponential time algorithm for this problem on maximum degree four graphs...

We present quantum graphs with remarkably regular spectral characteristics. We call them regular quantum graphs. Although regular quantum graphs are strongly chaotic in the classical limit, their quantum spectra are explicitly solvable in terms of periodic orbits. We present analytical solutions for the spectrum of regular quantum graphs in the form of explicit and exact periodic orbit expansio...