Reconstruction of complete interval tournaments

TL;DR

This paper provides a necessary and sufficient condition for reconstructing complete interval tournaments with given out-degree sequences, extending classical results and offering an efficient reconstruction algorithm.

Contribution

It generalizes existing theorems on tournament score sequences to multigraphs with bounded edge multiplicities and introduces a polynomial-time reconstruction algorithm.

Findings

Characterization of when a given out-degree vector can be realized

A reconstruction algorithm with worst-case complexity $O(bn^3)$

Extension of classical tournament theorems to multigraphs

Abstract

Let $a, b$ and $n$ be nonnegative integers $(b \geq a, \ b > 0, \ n \geq 1)$, $\mathcal{G}_n(a,b)$ be a multigraph on $n$ vertices in which any pair of vertices is connected with at least $a$ and at most $b$ edges and \textbf{v =} $(v_1, v_2, ..., v_n)$ be a vector containing $n$ nonnegative integers. We give a necessary and sufficient condition for the existence of such orientation of the edges of $\mathcal{G}_n(a,b)$, that the resulted out-degree vector equals to \textbf{v}. We describe a reconstruction algorithm. In worst case checking of \textbf{v} requires $\Theta(n)$ time and the reconstruction algorithm works in $O(bn^3)$ time. Theorems of H. G. Landau (1953) and J. W. Moon (1963) on the score sequences of tournaments are special cases $b = a = 1$ resp. $b = a \geq 1$ of our result.