We prove that for every number n ≥ 1, the n-iterated P3-path graph of G is isomorphic to G if and only if G is a collection of cycles, each of length at least 4. Hence, G is isomorphic to P3(G) if and only if G is a collection of cycles, each of length at least 4. Moreover, for k ≥ 4 we reduce the problem of characterizing graphs G such that Pk(G) ∼= G to graphs without cycles of length exceeding k.
Zero forcing number has recently become an interesting graph parameter studied in its own right since its introduction by the ''AIM Minimum Rank–Special Graphs Work Group'', whereas metric dimension is a well-known graph parameter. We investigate the metric dimension and the zero forcing number of some line graphs by first determining the metric dimension and the zero forcing number of the line graphs of wheel graphs and the bouquet of circles. We prove that Z(G) ≤ 2Z(L(G)) for a simple and connected graph G. Further, we show that Z(G) ≤ Z(L(G)) when G is a tree or when G contains a Hamiltonian path and has a certain number of edges. We compare the metric dimension with the zero forcing number of a line graph by demonstrating a couple of inequalities between the two parameters. We end by stating some open problems.
A graph $G$ is a minimal claw-free graph (m.c.f. graph) if it contains no $K_{1,3}$ (claw) as an induced subgraph and if, for each edge $e$ of $G$, $G-e$ contains an induced claw. We investigate properties of m.c.f. graphs, establish sharp bounds on their orders and the degrees of their vertices, and characterize graphs which have m.c.f. line graphs.
Let $G=(V(G),E(G))$ be a graph. Gould and Hynds (1999) showed a well-known characterization of $G$ by its line graph $L(G)$ that has a 2-factor. In this paper, by defining two operations, we present a characterization for a graph $G$ to have a 2-factor in its line graph $L(G).$ A graph $G$ is called $N^{2}$-locally connected if for every vertex $x\in V(G),$ $G[\{y\in V(G)\; 1\leq {\rm dist}_{G}(x,y)\leq 2\}]$ is connected. By applying the new characterization, we prove that every claw-free graph in which every edge lies on a cycle of length at most five and in which every vertex of degree two that lies on a triangle has two $N^{2}$-locally connected adjacent neighbors, has a $2$-factor. This result generalizes the previous results in papers: Li, Liu (1995) and Tian, Xiong, Niu (2012), and is the best possible.