# The phylogeny number in the aspect of triangles and diamonds of a graph

## TL;DR

This paper investigates the phylogeny number of graphs, especially those with triangles and diamonds, providing bounds and extending previous theorems, with implications for Bayesian networks and competition graphs.

## Contribution

It unifies existing theorems on phylogeny numbers for graphs with few triangles and extends bounds to graphs with many triangles, also answering a question by Wu et al.

## Key findings

- Derived bounds for phylogeny numbers in graphs with up to two triangles.
- Extended bounds to graphs with many triangles.
- Proved a key theorem impacting phylogeny number calculations.

## Abstract

Given an acyclic digraph $D$, the competition graph of $D$, denoted by $C(D)$, is the simple graph having vertex set $V(D)$ and edge set $\{uv \mid (u, w), (v, w) \in A(D) \text{ for some } w \in V(D) \}$. The phylogeny graph of an acyclic digraph $D$, denoted by $P(D)$, is the graph with the vertex set $V(D)$ and the edge set $E(U(D)) \cup E(C(D))$ where $U(D)$ denotes the underlying graph of $D$. The notion of phylogeny graphs was introduced by Roberts and Sheng~\cite{roberts1997phylogeny} as a variant of competition graph. Moral graphs having arisen from studying Bayesian networks are the same as phylogeny graphs. In this paper, we integrate the existing theorems computing phylogeny numbers of connected graph with a small number of triangles into one proposition: for a graph $G$ containing at most two triangle, $ |E(G)|-|V(G)|-2t(G)+d(G)+1 \le p(G) \le |E(G)|-|V(G)|-t(G)+1 $ where $t(G)$ and $d(G)$ denote the number of triangles and the number of diamond in $G$, respectively. Then we show that these inequalities hold for graphs with many triangles. In the process of showing it, we derive a useful theorem which plays a key role in deducing various meaningful results including a theorem that answers a question given by Wu~{\it et al.}~\cite{Wu2019}.