Related papers: Vertex elimination orderings for hereditary graph …

In a bounded max-coloring of a vertex/edge weighted graph, each color class is of cardinality at most $b$ and of weight equal to the weight of the heaviest vertex/edge in this class. The bounded max-vertex/edge-coloring problems ask for…

Classes of graphs with bounded expansion are a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r,…

A graph class is $\chi$-bounded if the only way to force large chromatic number in graphs from the class is by forming a large clique. In the 1970s, Erd\H{o}s conjectured that intersection graphs of straight-line segments in the plane are…

Vertex deletion problems for graphs are studied intensely in classical and parameterized complexity theory. They ask whether we can delete at most k vertices from an input graph such that the resulting graph has a certain property.…

There are many methods to find a maximum (or maximal) clique in large networks. Due to the nature of combinatorics, computation becomes exponentially expensive as the number of vertices in a graph increases. Thus, there is a need for…

We study the parameterized complexity of various classic vertex-deletion problems such as Odd cycle transversal, Vertex planarization, and Chordal vertex deletion under hybrid parameterizations. Existing FPT algorithms for these problems…

Many variations of the classical graph coloring model have been intensively studied due to their multiple applications; scheduling problems and aircraft assignments, for instance, motivate the robust coloring problem. This model gets to…

We consider hereditary classes of graphs equipped with a total order. We provide multiple equivalent characterisations of those classes which have bounded twin-width. In particular, we prove a grid theorem for classes of ordered graphs…

In the k-Apex problem the task is to find at most k vertices whose deletion makes the given graph planar. The graphs for which there exists a solution form a minor closed class of graphs, hence by the deep results of Robertson and Seymour,…

For a class of graphs $\mathcal{P}$, the Bounded $\mathcal{P}$-Block Vertex Deletion problem asks, given a graph $G$ on $n$ vertices and positive integers $k$ and $d$, whether there is a set $S$ of at most $k$ vertices such that each block…

Vertex Descent is a local search algorithm which forms the basis of a wide spectrum of tabu search, simulated annealing and hybrid evolutionary algorithms for graph colouring. These algorithms are usually treated as experimental and provide…

In a colouring of a graph, a vertex is b-chromatic if it is adjacent to a vertex of every other colour. We consider four well-studied colouring problems: b-Chromatic Number, Tight b-Chromatic Number, Fall Chromatic Number and Fall…

Treewidth is a parameter that emerged from the study of minor closed classes of graphs (i.e. classes closed under vertex and edge deletion, and edge contraction). It in some sense describes the global structure of a graph. Roughly, a graph…

A module of a graph G is a set of vertices that have the same set of neighbours outside. Modules of a graphs form a so-called partitive family and thereby can be represented by a unique tree MD(G), called the modular decomposition tree.…

It is well-known that the graph isomorphism problem can be posed as an equivalent problem of determining whether an auxiliary graph structure contains a clique of specific order. However, the algorithms that have been developed so far for…

We propose an unsupervised approach for learning vertex orderings for the maximum clique problem by framing it within a permutation-based framework. We transform the combinatorial constraints into geometric relationships such that the…

Brooks' Theorem [R. L. Brooks, On Colouring the Nodes of a Network, Proc. Cambridge Philos. Soc.} 37:194-197, 1941] states that every graph $G$ with maximum degree $\Delta$, has a vertex-colouring with $\Delta$ colours, unless $G$ is a…

We present new results on the classical algorithm of variable elimination, which underlies many algorithms including for probabilistic inference. The results relate to exploiting functional dependencies, allowing one to perform inference…

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. A graph $G$ is $k$-vertex-critical if every proper induced subgraph of $G$ has chromatic number less than $k$,…

We study the problem of graph clustering where the goal is to partition a graph into clusters, i.e. disjoint subsets of vertices, such that each cluster is well connected internally while sparsely connected to the rest of the graph. In…