The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer k ≥ 0, to delete at most k vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback verte...

We show complexity results for some generalizations of the graph coloring problem on two classes of perfect graphs, namely clique trees and unit interval graphs. We deal with the μ-coloring problem (upper bounds for the color on each vertex), the precoloring extension problem (a subset of vertices colored beforehand), and a problem generalizing both of them, the (γ, μ)-coloring problem (lower a...

Cliquewidth and NLC-width are two closely related parameters that measure the complexity of graphs. Both cliqueand NLC-width are defined to be the minimum number of labels required to create a labelled graph by certain terms of operations. Many hard problems on graphs become solvable in polynomial time if the inputs are restricted to graphs of bounded cliqueor NLC-width. Cliquewidth and NLC-wid...

An NP-complete coloring or homomorphism problem may become polynomial time solvable when restricted to graphs with degrees bounded by a small number, but remain NP-complete if the bound is higher. For instance, 3-colorability of graphs with degrees bounded by 3 can be decided by Brooks’ theorem, while for graphs with degrees bounded by 4, the 3-colorability problem is NP-complete. We investigat...

In a graph G, an efficient dominating set is a subset D of vertices such that D is an independent set and each vertex outside D has exactly one neighbor in D. The Minimum Weight Efficient Dominating Set (Min-WED) problem asks for an efficient dominating set of total minimum weight in a given vertex-weighted graph; the Maximum Weight Efficient Dominating Set (Max-WED) problem is defined similarl...

A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. A function is a pebbling threshold for a sequence of graphs if a randomly chosen configuration of asymptotically more pebbles is almost surely solvable, while one of asymptotically fewer pebbles is almost surely not. In this note we show that the...

In 1980 Piere Duchet conjectured that odd directed cycles are the only edge minimal kernel-less connected digraphs i.e. in which after the removal of any edge a kernel appears. Although this conjecture was disproved recently by Apartsin, Ferapontova and Gurvich (1996), the following modiication of Duchet's conjecture still holds: odd holes (i.e. odd non-directed chordless cycles of length 5 or ...

An equivalence between generalised restricted solid-on-solid (RSOS) models, associated with sets of graphs, and multi-colour loop models is established. As an application we consider solvable loop models and in this way obtain new solvable families of critical RSOS models. These families can all be classified by the Dynkin diagrams of the simply-laced Lie algebras. For one of the RSOS models, l...

Cliquewidth and NLC-width are two closely related parameters that measure the complexity of graphs. Both cliqueand NLC-width are defined to be the minimum number of labels required to create a labelled graph by certain terms of operations. Many hard problems on graphs become solvable in polynomial-time if the inputs are restricted to graphs of bounded cliqueorNLC-width. Cliquewidth andNLC-width...

An induced matching M in a graph G is dominating if every edge not in M shares exactly one vertex with an edge in M . The dominating induced matching problem (also known as efficient edge domination) asks whether a graph G contains a dominating induced matching. This problem is generally NP-complete, but polynomial-time solvable for graphs with some special properties. In particular, it is solv...