Original Graphs of Link Graphs
نویسنده
چکیده
Let l > 0 be an integer, and G be a graph without loops. An l-link of G is a walk of length l in which consecutive edges are different. We identify an l-link with its reverse sequence. The l-link graph Ll(G) of G is defined to have vertices the l-links of G, such that two vertices of Ll(G) are adjacent if their corresponding l-links are the initial and final subsequences of an (l + 1)link of G. A graph G is called an l-root of a graph H if Ll(G) ∼= H . For example, L0(G) ∼= G. And the 1-link graph of a simple graph is the line graph of that graph. Moreover, let H be a finite connected simple graph. Whitney’s isomorphism theorem (1932) states ifH has two connected nonnull simple 1-roots, then H ∼= K3, and the two 1-roots are isomorphic to K3 and K1,3 respectively. This paper investigates the l-roots of finite graphs. We show that every lroot is a certain combination of a finite minimal l-root and trees of bounded diameter. This transfers the study of l-roots into that of finite minimal l-roots. As a qualitative generalisation of Whitney’s isomorphism theorem, we bound from above the number, size, order and maximum degree of minimal l-roots of a finite graph. This work forms the basis for solving the recognition and determination problems for l-link graphs in our future papers. As a byproduct, we characterise the l-roots of some special graphs including cycles. Similar results are obtained for path graphs introduced by Broersma and Hoede (1989). G is an l-path root of a graph H if H is isomorphic to the l-path graph of G. We bound from above the number, size and order of minimal l-path roots of a finite graph.
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تاریخ انتشار 2015