Path graphs

Abstract

The concept of a line graph is generalized to that of a path graph. The path graph f,(G) of a graph G is obtained by representing the paths Pk in G by vertices and joining two vertices whenever the corresponding paths f k in G form a path f k + , or a cycle C,. f,-graphs are characterized and investigated on isomorphism and traversability. Trees and unicyclic graphs with hamiltonian /?,-graphs are characterized.

On the connectivity and restricted edge-connectivity of 3-arc graphs

“…There is a bijection between the edges of X(G) and those of the 2-path graph P 2 (G), which is defined to have vertices the paths of length two in G such that two vertices are adjacent if and only if the union of the corresponding paths is a path or a cycle of length three, see [9]. Since P 2 (G) is a spanning subgraph of the second iterated line graph L 2 (G) = L(L(G)) (see e.g.…”

Isomorphisms and traversability of directed path graphs

“…We refer to [5] for results related to similar questions concerning line (di)graphs, and to [3] for analogous results on P 3 -graphs of (undirected) graphs. In this section we shall characterize all digraphs for which → P 3 (D) ∼ = D, and we shall see that…”

“…In [3] path graphs were introduced as a generalization of line graphs of (undirected) graphs. In the next section we shall introduce an analogous concept for directed graphs.…”

An Optimal Algorithm to Detect Balancing in Common-edge Sigraph

“…. Broersma and Hoede [9] defined in general path graphs P k (G) of G for any positive integer k as follows: P k (G) has for its vertex-set the set P k G of all distinct paths in G having k vertices, and two vertices in P k (G) are adjacent if they represent two paths P, Q ∈ P k G whose union forms either a path P k+1 or a cycle C k in G. Some improvement of their paper was subsequently given by [27,7,28].…”

“…As in [9] (also see, [7]) by a sigraph S we mean a 2 graph G = (V, E) called the underlying graph of S and denoted by S u , in which each edge x carries a value s(x) ∈ {+1, −1} called its sign; an edge x is positive or negative according to whether s(x) = +1 or s(x) = −1. The set of positive edges of S is denoted by E + (S) and E − (S) = E(G) − E + (S) is the set of negative edges of S. Graphs themselves regarded as sigraphs in which every edge is positive.…”