Disjoint Compatibility Graph of Non-Crossing Matchings of Points in Convex Position

Abstract

Let X 2k be a set of 2k labeled points in convex position in the plane. We consider geometric non-intersecting straight-line perfect matchings of X 2k . Two such matchings, M and M ′ , are disjoint compatible if they do not have common edges, and no edge of M crosses an edge of M ′ . Denote by DCM k the graph whose vertices correspond to such matchings, and two vertices are adjacent if and only if the corresponding matchings are disjoint compatible. We show that for each k ≥ 9, the connected components of DCM k form exactly three isomorphism classes -namely, there is a certain number of isomorphic small components, a certain number of isomorphic medium components, and one big component. The number and the structure of small and medium components is determined precisely.

Packing 1-Plane Hamiltonian Cycles in Complete Geometric Graphs

“…Thus, C 1 has at least one boundary edge different from v i v j and v j v j+1 on P 1,2 . This proves (1).…”

“…It is often useful to restrict the subgraphs of G to a certain class or property. Among all subgraphs of K n , plane spanning trees, plane Hamiltonian cycles or paths, and plane perfect matchings, are of interest [1,2,3,7,18] i.e., one may look for the maximum number of these subgraphs that can be packed into K n . For instance, a long-standing open question is to determine if the edges of K n , where n is even, can be partitioned into n 2 plane spanning trees?…”

LATIN 2018: Theoretical Informatics

“…Indeed, width(e) equals the sum of widths of the edges in the upper chain of polygon P e . Since we do not use an edge of maximum width in this chain, we have width(e s ) ≤ 1 2 width(e) for every vertex s ∈ S e , as claimed. Now suppose that k ≥ log 2 n and width(e 1 ) ≥ 1 for some e 1 ∈ E k .…”

“…Transition graphs on other common plane geometric graphs have been considered in the literature, but they do not allow for such a rich variety of operations. For the space of noncrossing matchings on S, a compatible exchange operation has been defined, but the transition graph is disconnected even if S is in convex position [1]; it is known that the transition graph has no isolated vertices [24]. Connectedness is known for bipartite geometric matchings, with a tight linear diameter bound [7].…”

“…Denote by L i the set of good leaves removed in iteration i. Each iteration removes at least a 1 24 -fraction of the vertices. After r ∈ O(log n) iterations, we are left with a single vertex p. The tree T 3 ∈ T (S) is the tree obtained by these "removal" operations.…”