The edge-closure of a claw-free graph is the line graph of a multigraph
نویسنده
چکیده
Ryjáček introduced a closure concept in claw-free graphs based on local completion at a locally connected vertex. He showed that the closure of a graph is the line graph of a triangle-free graph. Brousek and Holub gave an analogous closure concept of claw-free graphs, called the edge-closure, based on local completion at a locally connected edge. In this paper, it is shown that the edge-closure is the line graph of a multigraph.
منابع مشابه
A Closure for 1-Hamilton-Connectedness in Claw-Free Graphs
A graph G is 1-Hamilton-connected if G − x is Hamilton-connected for every vertex x ∈ V (G). In the paper we introduce a closure concept for 1-Hamiltonconnectedness in claw-free graphs. If G is a (new) closure of a claw-free graph G, then G is 1-Hamilton-connected if and only if G is 1-Hamilton-connected, G is the line graph of a multigraph, and for some x ∈ V (G), G − x is the line graph of a ...
متن کاملCollapsible graphs and Hamiltonian connectedness of line graphs
Thomassen conjectured that every 4-connected line graph is Hamiltonian. Chen and Lai [Z.-H. Chen, H.-J. Lai, Reduction techniques for super-Eulerian graphs and related topics— an update, in: Ku Tung-Hsin (Ed.), Combinatorics and Graph Theory, vol. 95, World Scientific, Singapore/London, 1995, pp. 53–69, Conjecture 8.6] conjectured that every 3-edge connected, essentially 6-edge connected graph ...
متن کاملClosure, clique covering and degree conditions for Hamilton-connectedness in claw-free graphs
We strengthen the closure concept for Hamilton-connectedness in claw-free graphs, introduced by the second and fourth authors, such that the strong closure GM of a claw-free graph G is the line graph of a multigraph containing at most two triangles or at most one double edge. Using the concept of strong closure, we prove that a 3-connected claw-free graph G is Hamilton-connected if G satisfies ...
متن کاملLine graphs of multigraphs and Hamilton-connectedness of claw-free graphs
We introduce a closure concept that turns a claw-free graph into the line graph of a multigraph while preserving its (non-)Hamiltonconnectedness. As an application, we show that every 7-connected claw-free graph is Hamilton-connected, and we show that the wellknown conjecture by Matthews and Sumner (every 4-connected clawfree graph is hamiltonian) is equivalent with the statement that every 4-c...
متن کاملLine completion number of grid graph Pn × Pm
The concept of super line graph was introduced in the year 1995 by Bagga, Beineke and Varma. Given a graph with at least r edges, the super line graph of index r, Lr(G), has as its vertices the sets of r-edges of G, with two adjacent if there is an edge in one set adjacent to an edge in the other set. The line completion number lc(G) of a graph G is the least positive integer r for which Lr(G) ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Ars Comb.
دوره 106 شماره
صفحات -
تاریخ انتشار 2012