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

The phylogeny graph of a digraph $D$, denoted by $P\left(D\right)$, has the vertex set $V\left(D\right)$ and an edge $uv$ if and only if $(u,v)$ or $(v,u)$ is an arc of $D$ or $u$ and $v$ have a common out-neighbor in $D$. The notion of phylogeny graphs was introduced by Roberts and Sheng (1997) as a variant of competition graph. Moral graphs having arisen from studying Bayesian networks are the same as phylogeny graphs. Any acyclic digraph $D$ for which $G$ is an induced subgraph of $P\left(D\right)$ and such that $D$ has no arcs from vertices outside of $G$ to vertices in $G$ is called a phylogeny digraph for $G$.

The phylogeny number $p\left(G\right)$ of $G$ is the smallest $r$ so that $G$ has a phylogeny digraph $D$ with $|V\left(D\right)\setminus V\left(G\right)|=r$. 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, $\left|E\left(G\right)\right|-\left|V\left(G\right)\right|-2t\left(G\right)+d\left(G\right)+1\le p\left(G\right)\le \left|E\left(G\right)\right|-\left|V\left(G\right)\right|-t\left(G\right)+1$ where $t\left(G\right)$ and $d\left(G\right)$ denote the number of triangles and the number of diamonds 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 et al. (2019).