A conformally indecomposable flow f on a signed graph Σ is a nonzero integral flow that cannot be decomposed into f = f1 + f2, where f1, f2 are nonzero integral flows having the same sign (both ≥ 0 or both ≤ 0) at every edge. This paper is to classify at integer scale conformally indecomposable flows into characteristic vectors of Eulerian cycle-trees — a class of signed graphs having a kind of...

A tree is called a caterpillar if all its leaves are adjacent to the same its path, and the path is called a spine of the caterpillar. Broersma and Tuinstra proved that if a connected graph G satisfies σ2(G) ≥ |G| − k + 1 for an integer k ≥ 2, then G has a spanning tree having at most k leaves. In this paper we improve this result as follows. If a connected graph G satisfies σ2(G) ≥ |G|−k+1 and...

In this paper, we introduce a concise representation, called right-distance sequences (or RD-sequences for short), to describe all t-ary trees with n internal nodes. A result reveals that there exists a close relationship between the representation and the well-formated integer sequences suggested by Zaks in 1980. Using recursion tree and its concomitant tables, a systematical way can help us t...

In metabolomics and other fields dealing with small compounds, mass spectrometry is applied as a sensitive high-throughput technique. Recently, fragmentation trees have been proposed to automatically analyze the fragmentation mass spectra recorded by such instruments. Computationally, this leads to the problem of finding a maximum weight subtree in an edge-weighted and vertex-colored graph, suc...

The polyhedral structure of the K-median problem on a tree is examined. Even for very small connected graphs, we show that additional constraints are needed to describe the integer polytope. We completely characterize those trees for which an optimal integer LP solution is guaranteed to exist. Also, a new and simpler proof of the worst case bound of an LP solution is provided. We find a new cla...

The suux tree of a string is the fundamental data structure of combinatorial pattern matching. Weiner Wei73], who introduced the data structure, gave an O(n) time algorithm algorithm for building the suux tree of an n character string drawn from a constant size alphabet. In the comparison model, there is a trivial (n log n) time lower bound based on sorting, and Weiner's algorithm matches this ...

A. We use the group (Z2,+) and two associated automorphisms, τ0, τ1, to generate all distinct, non-zero pairs of coprime, positive integers which we describe within the context of a binary tree which we denote T . While this idea is related to the Stern-Brocot tree and the map of relatively prime pairs, the parents of an integer pair these trees do not necessarily correspond to the paren...

The polyhedral structure of the K-median problem on a tree is examined. Even for very small connected graphs, we show that additional constraints are needed to describe the integer polytope. A complete description is given of those trees for which an optimal integer LP solution is guaranteed to exist. We present a new and simpler demonstration that an LP characterization of the 2-median problem...

Given a degree set D = {a1 < a2 < . . . < an} of nonnegative integers, the minimum number of vertices in any tree realizing the set D is known[10]. In this paper, we study the number of vertices and multiplicity of distinct degrees as parameters of tree realizations of degree sets. We explore this in the context of both directed and undirected trees and asymmetric directed graphs (graphs which ...

Let G be a simple graph with non-negative edge weights. Determining a minimum weight spanning tree is a fundamental problem that arises in network design and as a subproblem in many combinatorial optimization problems such as vehicle routing. In some applications, it is necessary to restrict the diameter of the spanning tree and thus one is interested in the problem : Find, in a given weighted ...