Counting Tree-Like Multigraphs with a Given Number of Vertices and Multiple Edges

TL;DR

The paper addresses counting non-isomorphic tree-like multigraphs with $n$ vertices and $Δ$ multiple edges, a problem with applications in chemical graph enumeration. It introduces a dynamic programming framework that roots graphs at a unicentroid or bicentroid and uses a canonical representation based on maximal descendant subgraphs to avoid duplicate counts. Recursive relations are derived over aFamily of subproblems $M(n,Δ,k≤,d≤,ℓ≤)$ and related equality variants, with explicit initial conditions, enabling a DP algorithm whose time and space complexities scale as $O(n^2(n + Δ(n + Δ^2 \,min\{n,Δ\})))$ and $O(n^2(Δ^3+1))$, respectively. Experimental results demonstrate counting efficiency up to $n=170$ and $Δ=50$ within about $930$ seconds, illustrating the method’s practicality for exploring the chemical graph space and informing future extensions to include self-loops and generation of tree-like multigraphs.

Abstract

The enumeration of chemical graphs is an important topic in cheminformatics and bioinformatics, particularly in the discovery of novel drugs. These graphs are typically either tree-like multigraphs or composed of tree-like multigraphs connected to a core structure. In both cases, the tree-like components play a significant role in determining the properties and activities of chemical compounds. This paper introduces a method based on dynamic programming to efficiently count tree-like multigraphs with a given number $n$ of vertices and $Δ$ multiple edges. The idea of our method is to consider multigraphs as rooted multigraphs by selecting their unicentroid or bicentroid as the root, and define their canonical representation based on maximal subgraphs rooted at the children of the root. This representation guarantees that our proposed method will not repeat a multigraph in the counting process. Finally, recursive relations are derived based on the number of vertices and multiple edges in the maximal subgraphs rooted at the children of roots. These relations lead to an algorithm with a time complexity of $\mathcal{O}(n^2(n + Δ(n + Δ^2 \cdot \min\{n, Δ\})))$ and a space complexity of $\mathcal{O}(n^2(Δ^3+1))$. Experimental results show that the proposed algorithm efficiently counts the desired multigraphs with up to 170 vertices and 50 multiple edges in approximately 930 seconds, confirming its effectiveness and potential as a valuable tool for exploring the chemical graph space in novel drug discovery.

Figures (13)

• Figure 1: Chemical structures of compounds from PubChem database: (a) Aspartic Acid C$_{4}$H$_{7}$NO$_{4}$, a tree-like multigraph, PubChem CID: 5960; (b) Homovanillic Acid C$_{9}$H$_{10}$O$_{4}$, which includes a ring structure, Pubchem CID: 1738.
• Figure 2: Illustration of a multigraph which is a 2D graphical structure of Homovanillic Acid shown in Fig. \ref{['fig:peptide']}(b).
• Figure 3: Rooted tree-like multigraph: (a) 2D graphical structure of Aspartic Acid shown in Fig. \ref{['fig:peptide']}(a); (b) The multigraph with a specified root $r$. For vertex $v_3$, $v_4$ is its parent, $v_1$ and $v_2$ are its children. Vertex $v_3$ has no sibling, while $v_4$ and $r$ are its ancestors, and $v_1$ and $v_2$ are its descendants.
• Figure 4: (a) Unicentroid $v_0$ in $T_1$ and $v_1$ in $T_2$; (b) Bicentroid edge formed by vertex $v_1$ and $v_2$ in $T_3$.
• Figure 5: Illustration of rooted tree-like multigraphs $M$ and $N$, which are isomorphic due to the mapping $\varphi$ such that $\varphi(r_M)=r_N$, $\varphi(v_1)=v_3$, $\varphi(v_2)=v_1$, $\varphi(v_3)=v_2$, $\varphi(v_4)=v_7$, $\varphi(v_5)=v_8$, $\varphi(v_6)=v_4$, $\varphi(v_7)=v_6$ and $\varphi(v_8)=v_5$,.

Theorems & Definitions (17)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Lemma 3
• proof
• Lemma 4
• proof