A Fast Algorithm for Computing Irreducible Triangulations of Closed Surfaces in E and Its Application to the TriQuad Problem

Let S be a compact surface with empty boundary. A classical result from the 1920s by Tibor Radó asserts that every compact surface with empty boundary (usually called a closed surface) admits a triangulation [1]. Let e be any edge of a triangulation T of S . The contraction of e in T consists of contracting e to a single vertex and collapsing each of the two triangles meeting e into a single edge. If the result of contracting e in T is still a triangulation of S , then e is said to be contractible; else it is non-contractible. A triangulation T of S is said to be irreducible if and only if every edge of T is non-contractible. Barnette and Edelson [2] showed that all closed surfaces have finitely many irreducible surfaces. More recently, Boulch, de Verdière, and Nakamoto [3] showed the same result for compact surfaces with a nonempty boundary. Irreducible triangulations have proved to be an important tool for tackling problems in combinatorial topology, and discrete and computational geometry. The reasons are two-fold. First, all irreducible triangulations of any given compact surface form a “basis” for all triangulations of the same surface. Indeed, every triangulation of the surface can be obtained from at least one of its irreducible triangulations by a sequence of vertex splittings [4, 5], where the vertex splitting operation is the inverse of the edge contraction operation. Second, some problems on triangulations can be solved by considering irreducible triangulations only. In particular, irreducible triangulations have been used for proving the existence of geometric realizations (in some E) of triangulations of certain surfaces, where E is the d-dimensional Euclidean space [6, 7], for studying properties of diagonal flips on surface triangulations [8, 9, 10, 11], for characterizing the structure of flexible triangulations of the projective plane [12], and for finding lower and upper bounds for the maximum number of cliques in an n-vertex graph embeddable in a given surface [13]. An irreducible triangulation is also “small”, as its number of vertices is at most linear in the genus of the surface [14, 3]. However, the number of vertices of all irreducible triangulations of the same surface may vary, while any irreducible triangulation of smallest size (known as minimal) has Θ( √ g) vertices if the genus g of the surface is positive [15]. The sphere has a unique irreducible triangulation, which is the boundary of a tetrahedron [16]. The torus has exactly 21 irreducible triangulations, whose numbers of vertices vary from 7 to 10 [17]. The projective plane has only two irreducible triangulations, one with 6 vertices and the other with 7 vertices [18]. The Klein bottle has exactly 29 irreducible triangulations with numbers of vertices ranging from 8 to 11 [19]. Sulanke devised and implemented an algorithm for generating all irreducible triangulations of compact surfaces with empty boundary [5]. Using this algorithm, Sulanke rediscovered the aforementioned irreducible triangulations and generated the complete sets of irreducible triangulations of the double torus, the triple cross surface, and the