Blockers for Simple Hamiltonian Paths in Convex Geometric Graphs of Even Order

Abstract

Let G be a complete convex geometric graph on 2m vertices, and let F be a family of subgraphs of G. A blocker for F is a set of edges, of smallest possible size, that meets every element of F . In [3] we gave an explicit description of all blockers for the family of simple perfect matchings (SPMs) of G. In this paper we show that the family of simple Hamiltonian paths (SHPs) in G has exactly the same blockers as the family of SPMs. Our argument is rather short, and provides a much simpler proof of the result of [3].

Blockers for Simple Hamiltonian Paths in Convex Geometric Graphs of Odd Order

“…See [7] for the exact definition of caterpillars.) In [4] we showed that the blockers for the family H of simple Hamiltonian paths (SHPs) in CK(2m) are exactly the same as the blockers for SPMs.…”

“…The next step of the proof will be to split CK(2m − 1) into two complete convex geometric graphs G + and G − of even order, leaving out just one vertex, in such a way that the edges of B(ǫ) in G + [resp., G − ] form a blocker for SHP's in G + [resp., G − ]. Here we use the sufficiency part of the characterization of blockers for SHP's in complete convex geometric graphs of even order, given in [4]. To finalize the proof, we shall use the part of P that runs through G + [or G − ] to construct an SHP in G + [or G − ] that avoids a blocker for SHP's in G + [or G − ], and thus reach a contradiction.…”

“…

Let G be a complete convex geometric graph, and let F be a family of subgraphs of G. A blocker for F is a set of edges, of smallest possible size, that has an edge in common with every element of F . In [4] we gave an explicit description of all blockers for the family of simple (i.e., non-crossing) Hamiltonian paths (SHPs) in G in the 'even' case |V (G)| = 2m. It turned out that all the blockers are simple caterpillar trees of a certain class.

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Blockers for Triangulations of a Convex Polygon and a Geometric Maker-Breaker Game

“…The most satisfactory answer for the 'blockers' question is not only determining their size, but rather giving a complete characterization of the set of blockers. Such a characterization has been obtained for quite a few families of simple (i.e., non-crossing) graphs, including the family of all simple perfect matchings in [16], the family of all simple spanning trees in [12], the family of all Hamiltonian paths in [18], etc. The characterizations gave rise to interesting classes of examples, including caterpillar graphs (see [10,16]), combs (see [19]) and semi-simple perfect matchings (see [17]), and had applications to the structure of the 'flip graphs' of the respective structures (see [12,13]).…”

Blocking dominating sets for $H$-free graphs via edge contractions

“…Such a designation follows from the fact that the set of vertices or edges involved can be viewed as "blocking" the parameter π. Identifying such sets may provide information on the structure of the input graph; for instance, if π = α, k = d = 1 and O = {vertex deletion}, the problem is equivalent to testing whether the input graph contains a vertex that is in every maximum independent set (see [18]). Blocker problems have received much attention in the recent literature (see for instance [1,2,3,4,5,7,8,9,11,12,13,15,16,17,18,19]) and have been related to other well-known graph problems such as Hadwiger Number, Club Contraction and several graph transversal problems (see for instance [7,17]). The graph parameters considered so far in the literature are the chromatic number, the independence number, the clique number, the matching number and the vertex cover number while the set O is a singleton consisting of a vertex deletion, edge contraction, edge deletion or edge addition.…”