In this paper we continue the study of the edge intersection graphs of single bend paths on a rectangular grid (i.e., the edge intersection graphs where each vertex is represented by one of the following shapes: x, p, y, q). These graphs, called B1-EPG graphs, were first introduced by Golumbic et al (2009) [5]. We consider the natural subclasses of B1-EPG formed by the subsets of the four singl...

Deciding whether a collection of unrooted trees is compatible is a fundamental problem in phylogenetics. Two different graph-theoretic characterizations of tree compatibility have recently been proposed. In one of these, tree compatibility is characterized in terms of the existence of a specific kind of triangulation in a structure known as the display graph. An alternative characterization exp...

Nanomedicine is a cutting-edge field at the intersection of nanotechnology and medicine that has experienced significant advancements in recent decades [...]

Let L3 be the class of edge intersection graphs of linear 3-uniform hypergraphs. The problem of recognizing G ∈ L3 is NP-complete. Denote by δALG the minimal integer such that the problem ”G ∈ L3 ” is polynomially solvable in the class of graphs G with the minimal vertex degree δ(G) ≥ δALG and by δFIS the minimal integer such that L3 can be characterized by a finite list of forbidden induced su...

The class of edge intersection graphs of a collection of paths in a tree (EPT graphs) is investigated, where two paths edge intersect if they share an edge. The cliques of an EPT graph are characterized and shown to have strong Helly number 4. From this it is demonstrated that the problem of finding a maximum clique of an EPT graph can be solved in polynomial time. It is shown that the strong p...

We investigate edge-intersection graphs of paths in the plane grid regarding a parameter called the bend-number. The bend-number is related to the interval-number and the track-number of a graph. We provide new upper and lower bounds of the bend-number of any given simple graph in terms of the coloring number, edge clique covers and the maximum degree. We show that the bend-number of an outerpl...

Let T = (V(T), E(T)) be a tree and B = {Pi} denote a collection of non-trivial (i.e., of length at least 1) simple paths in T, where a path P= (v,, u2, . . . , Q) is considered in the sequel as the collection {{uJ, {Q}, . . . ,{z)J, {u,, u2}, {u2, I+}, _ . . , {w,-,, Q}}. The intersection graph 0(S, 9) of 9 over set S has vertices which correspond to the paths in 9 and two vertices uk and U! ar...

Let R be monomial sub-algebra of k[x1, . . . , xN ] generated by square free monomials of degree two. This paper addresses the following question: when is R a complete intersection? For such a k-algebra we can associate a graph G whose vertices are x1, . . . , xN and whose edges are {(xi, xj)|xixj ∈ R}. Conversely, for any graph G with vertices {x1, . . . , xN} we define the edge algebra associ...

In this paper, we study the Hankel edge ideals of graphs. We determine minimal prime ideal labeled Hamiltonian and semi-Hamiltonian graphs, investigate radicality, being a complete intersection, almost intersection set-theoretic for such also consider trees with natural labeling, called rooted labeling. characterize whose is moreover, those initial respect to reverse lexicographic order satisfi...

How fast is electronic relaxation? Technical advances in extending spectroscopy to the attosecond time scale has caused considerable interest experimental studies of various ultrafast processes that were previously inaccessible. Using transient absorption around carbon K-edge accompanied by ab initio quantum dynamics simulations, Zinchenko et al. investigated ultrafast, non-adiabatic ethylene c...