Abstract Addressing a quest by Gupta et al. (in: Proceedings of the 41st international colloquium on automata, languages, and programming (ICALP 2014), vol 8572 LNCS. Springer, pp 563–575, we provide first, comprehensive study finding short s – t path in multistage graph model, referred to as Multistage Path problem. Herein, given sequence graphs over same vertex set but changing edge sets, tas...

The path-table P(T ) of a tree T collects information regarding the paths in T: for each vertex v, the row of P(T ) relative to v lists the number of paths containing v of the various lengths. We call this row the path-row of v in T. Two trees having the same path-table (up to reordering the rows) are called path-congruent (or path-isomorphic). Motivated byKelly–Ulam’sReconstructionConjecture a...

When a real-valued function of one variable is approximated by its n th degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue p-norms in cases where f (n) or f (n+1) are Henstock–Kurzweil integrable. When the only assumption is that f (n) is Henstock–Kurzweil integrable then a modified form of the n th degree Taylor polynomial is used. When the only assumption i...

Let G = (V,E) (|V | = n and |E| = m) be an undirected graph with positive edge weights. Let PG(s, t) be a shortest s− t path in G. Let l be the number of edges in PG(s, t). The Edge Replacement Path problem is to compute a shortest s− t path in G\{e}, for every edge e in PG(s, t). The Node Replacement Path problem is to compute a shortest s− t path in G\{v}, for every vertex v in PG(s, t). In t...

Let G be a graph and T a set of vertices. A T -path in G is a path that begins and ends in T , and none of its internal vertices are contained in T . We define a T -path covering to be a union of vertex-disjoint T paths spanning all of T . Concentrating on graphs that are tough (the removal of any nonempty set X of vertices yields at most |X| components), we completely characterize the edges th...

We introduce a new family of graphs for which the Hamiltonian path problem is non-trivial and yet has a linear time solution. The square of a graph G = (V, E), denoted as G, is a graph with the set of vertices V , in which two vertices are connected by an edge if there exists a path of length at most 2 connecting them in G. Harary & Schwenk (1971) proved that the square of a tree T contains a H...

disordered t-shaped graphene nanodevice (tgn) was designed and studied in this paper. we demonstrated the intrinsic transport properties of the tgn by using landauer approach. knowing the transmission probability of an electron the current through the system is obtained using landauer-buttiker formalism. the effects of single disorder on conductance, current and on the transport length scales a...

Given a polygonal path P with vertices p1, p2, . . . , pn ∈ Rd and a real number t ≥ 1, a path Q = (pi1 , pi2 , . . . , pik) is a t-distance-preserving approximation of P if 1 = i1 < i2 < . . . < ik = n and each straight-line edge (pij , pij+1) of Q approximates the distance between pij and pij+1 along the path P within a factor of t. We present exact and approximation algorithms that compute s...