A graph G on n vertices is pancyclic if G contains cycles of all lengths`for 3 ` n and G is cycle extendable if for every nonhamiltonian cycle C G there is a cycle C 0 G such that V (C) V (C 0) and jV (C 0) n V (C)j = 1. We prove that (i) every 2-connected K 1;3-free graph is pancyclic, if G is P 5-free and n 6, if G is P 6-free and n 10, or if G is P 7-free, deer-free and n 14, and (ii) every ...

A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of (distinct) vertices of A intersect. In other words, A is d-scattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special cl...

In this paper we propose a structural characterization for a class of bipartite graphs defined by two forbidden induced subgraphs. We show that the obtained characterization leads to polynomial-time algorithms for several problems that are NP-hard in general bipartite graphs.

An antimatroid is an accessible union-closed family of subsets of a 0nite set. A number of classes of antimatroids are closed under taking minors such as point-search antimatroids of rooted (di)graphs, line-search antimatroids of rooted (di)graphs, shelling antimatroids of rooted trees, shelling antimatroids of posets, etc. The forbidden minor characterizations are known for point-search antima...

Let H be a set of connected graphs. A graph G is said to be H-free if G does not contain any element of H as an induced subgraph. Let Fk(H) be the set of k-connected H-free graphs. When we study the relationship between forbidden subgraphs and a certain graph property, we often allow a finite exceptional set of graphs. But if the symmetric difference of Fk(H1) and Fk(H2) is finite and we allow ...

We prove that every 2{connected K 1;3-free and Z 3 ?free graph is hamiltonian except for two graphs. Furthermore, we give a complete characterization of all 2?connected, K 1;3-free graphs, which are not pancyclic, and which are Z 3-free, B-free, W-free, or HP 7 ?free.

We introduce a notion of bipartite minors and prove a bipartite analog of Wagner’s theorem: a bipartite graph is planar if and only if it does not contain K3,3 as a bipartite minor. Similarly, we provide a forbidden minor characterization for outerplanar graphs and forests. We then establish a recursive characterization of bipartite (2, 2)-Laman graphs — a certain family of graphs that contains...

A graph is equimatchable if all of its maximal matchings have the same size. Equimatchable graphs are extensively studied in the literature mainly from structural point of view. Here we provide, to the best of our knowledge, the first family of forbidden subgraphs of equimatchable graphs. Since equimatchable graphs are not hereditary, the task of finding forbidden subgraphs requires the use of ...

We analyse the effects of predation in a vertical differentiation model, where the highquality incumbent is able to price discriminate while the low-quality entrant sets a uniform price. The incumbent may act as a predator, that is, it may price below its marginal costs on a subset of consumers to induce the rival’s exit. We show that the entrant may adopt an aggressive attitude to make predati...

A common task in phylogenetics is to find an evolutionary tree representing proximity relationships between species. This motivates the notion of leaf powers: a graph G = (V,E) is a leaf power if there exist a tree T on leafset V and a threshold k such that uv ∈ E if and only if the distance between u and v in T is at most k. Characterizing leaf powers is a challenging open problem, along with ...