An end of a graph G is an equivalence class rays, where two rays are equivalent if there infinitely many vertex-disjoint paths between them in G. The degree the maximum cardinality collection pairwise disjoint this class.

Disjoint paths have applications in establishing bottleneck-free communication between processors in a network. The problem of finding minimum delay disjoint paths in a network directly reduces to the problem of finding the minimal disjoint paths in the graph which models the network. Previous results for this problem on chordal graphs were an O(| V | | E |) algorithm for 2 edge disjoint paths ...

An n-bit Gray code is a sequence of all n-bit vectors such that consecutive vectors differ in a single bit. It is well-known that given α, β ∈ {0, 1}, an n-bit Gray code between α and β exists iff the Hamming distance d(α, β) of α and β is odd. We generalize this classical result to k pairwise disjoint pairs αi, βi ∈ {0, 1}: if d(αi, βi) is odd for all i and k < n, then the set of all n-bit vec...

An area based counterpart of the binary structural opening spectra is developed. It is shown that these area opening and closing spectra can be computed using an adaptation of Tarjan’s union-find algorithm. These spectra provide rotation, translation, and scale invariant pattern vectors for texture analysis.

Drawing together techniques from combinatorics and computer science, we improve the census algorithm for enumerating closed minimal P-irreducible 3-manifold triangulations. In particular, new constraints are proven for face pairing graphs, and pruning techniques are improved using a modification of the union-find algorithm. Using these results we catalogue all 136 closed nonorientable P-irreduc...

Many applications of congruence closure nowadays require the ability of recovering, among the thousands of input equations, the small subset that caused the equivalence of a given pair of terms. For this purpose, here we introduce an incremental congruence closure algorithm that has an additional Explain operation. First, two variations of union-find data structures with Explain are introduced....

For a graph G = ( V , E ) and set S ⊆ of size at least 2, an S-Steiner tree T is subgraph that with . Two -Steiner trees ′ are internally disjoint (resp. edge-disjoint if ∩ ∅ ). Let κ λ denote the maximum number edge-disjoint) in The k-tree connectivity k edge-connectivity then defined as minimum ), where ranges over all -subsets In Li et al. (2018) [12] authors conjectured connected has vertic...

We prove that every disjoint NP-pair is polynomial-time, many-one equivalent to the canonical disjoint NP-pair of some propositional proof system. Therefore, the degree structure of the class of disjoint NP-pairs and of all canonical pairs is identical. Secondly, we show that this degree structure is not superficial: Assuming there exist P-inseparable disjoint pairs, there exist intermediate di...

Disjoint NP-pairs are a well studied complexity-theoretic concept with important applications in cryptography and propositional proof complexity. In this paper we introduce a natural generalization of the notion of disjoint NP-pairs to disjoint k-tuples of NP-sets for k ≥ 2. We define subclasses of the class of all disjoint k-tuples of NP-sets. These subclasses are associated with a proposition...