A graph is polar if the vertex set can be partitioned into A and B in such a way that A induces a complete multipartite graph and B induces a disjoint union of cliques (i.e., the complement of a complete multipartite graph). Polar graphs naturally generalize several classes of graphs such as bipartite graphs, cobipartite graphs and split graphs. Recognizing polar graphs is an NP-complete proble...

After investigating the ball-packability of some small graphs, we give a full characterisation, in terms of forbidden induced subgraphs, for the stacked 4-polytopes whose 1-skeletons can be realised by the tangency relations of a ball packing.

Background The conventional MRI protocol for the characterization of atherosclerotic plaques involves a series of scans that provide multiple contrast weightings for resolving highrisk plaque characteristics [1]. However, mis-registration between image sets due to the inter-scan motion often compromises evaluation accuracy and image interpretation is expertise dependent [2]. The aim of this wor...

In the traveling salesman path problem, we are given a set of cities, traveling costs between city pairs and fixed source and destination cities. The objective is to find a minimum cost path from the source to destination visiting all cities exactly once. In this paper, we study polyhedral and combinatorial properties of a variant we call the traveling salesman walk problem, in which the object...

We provide a doubly exponential upper bound in p on the size of forbidden pivot-minors for symmetric or skew-symmetric matrices over a fixed finite field F of linear rank-width at most p. As a corollary, we obtain a doubly exponential upper bound in p on the size of forbidden vertex-minors for graphs of linear rank-width at most p. This solves an open question raised by Jeong, Kwon, and Oum [Ex...

Forbidden characterizations may sometimes be the most natural way to describe families of graphs, and yet these characterizations are usually very hard to exploit for enumerative purposes. By building on the work of Gioan and Paul (2012) and Chauve et al. (2014), we show a methodology by which we constrain a split-decomposition tree to avoid certain patterns, thereby avoiding the corresponding ...

In this paper, we consider pairs of forbidden subgraphs that imply the existence of a 2-factor in a graph. For d ≥ 2, let Gd be the set of connected graphs of minimum degree at least d. Let F1 and F2 be connected graphs and let H be a set of connected graphs. Then {F1, F2} is said to be a forbidden pair for H if every {F1, F2}-free graph in H of sufficiently large order has a 2-factor. Faudree,...

An upper dominating set in a graph is a minimal (with respect to set inclusion) dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard, but can be solved in polynomial time in some restricted graph classes, such as P4-free graphs or 2K2-free graphs. For classes defined by finitely many forbidden induced subgraphs, the boundary separating diffi...

In an edge deletion problem one is asked to delete at most k edges from a given graph such that the resulting graph satisfies a certain property. In this work, we study four NP-complete edge deletion problems where the goal graph has to be a chain, a split, a threshold, or a co-trivially perfect graph, respectively. All these four graph classes are characterized by a common forbidden induced su...

The most famous result of extremal combinatorics is probably the celebrated theorem of Turán [20] determining the maximal number ex(n;Kr) of edges in a Kr-free graph on n vertices. Asymptotically, ex(n;Kr) ≈ ( 1− 1 r−1 ) ( n 2 ) . The non-trivial part (that is, the upper bound) of this theorem in the contrapositive form can be stated as follows: any graph G with m > ex(n;Kr) edges contains at l...