Number of Subgraphs and Their Converses in Tournaments and New Digraph Polynomials

TL;DR

This work introduces a digraph polynomial $P_D(x)$ as a necessary condition for converse invariance of an orgraph $D$ in tournaments, proving that $P_D(x)=P_{-D}(x)$ whenever $D$ is converse invariant. Using a probabilistic construction with a biased tournament extension, the authors derive structural consequences and parity-type constraints on degree sequences, enabling a partial characterization: all orientations of trees with diameter at most $3$ are classified, and many orientations of graphs with $\Delta\ge3$ are shown not to be converse invariant. Beyond these results, the paper develops recursive construction techniques—vertex-transitive bridge-additions and bridge-mirroring—that generate infinite families of converse invariant orgraphs not isomorphic to their converse, including nontrivial examples with the same property. The combined polynomial- and construction-based approach yields both concrete classifications and broad non-invariance results, and it leads to a guiding conjecture that recasts converse invariance in terms of $D\cong -D$ or bridge-mirroring from a path, with potential implications for the understanding of subgraph counts in tournaments and related digraph polynomials.

Abstract

An oriented graph $D$ is converse invariant if, for any tournament $T$, the number of copies of $D$ in $T$ is equal to that of its converse $-D$. El Sahili and Ghazo Hanna [J. Graph Theory 102 (2023), 684-701] showed that any oriented graph $D$ with maximum degree at most 2 is converse invariant. They proposed a question: Can we characterize all converse invariant oriented graphs? In this paper, we introduce a digraph polynomial and employ it to give a necessary condition for an oriented graph to be converse invariant. This polynomial serves as a cornerstone in proving all the results presented in this paper. In particular, we characterize all orientations of trees with diameter at most 3 that are converse invariant. We also show that all orientations of regular graphs are not converse invariant if $D$ and $-D$ have different degree sequences. In addition, in contrast to the findings of El Sahili and Ghazo Hanna, we prove that every connected graph $G$ with maximum degree at least $3$, admits an orientation $D$ of $G$ such that $D$ is not converse invariant. We pose one conjecture.

Figures (2)

• Figure 1: $D+(u,v)$
• Figure 2: $2D^+_u$

Theorems & Definitions (20)

• Theorem 1.1: SH2023
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.3
• proof
• Corollary 2.4
• proof
• Corollary 3.1
• Theorem 3.2