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Global Least Common Ancestor (LCA) Networks

Anna Lindeberg, Bruno J. Schmidt, Manoj Changat, Ameera Vaheeda Shanavas, Peter F. Stadler, Marc Hellmuth

TL;DR

This work characterizes global LCA-DAGs as exactly the networks whose reachability poset $(V(G),\preceq_G)$ forms a join-semilattice, linking LCA uniqueness to semilattice structure and to topological-minor conditions. It provides a constructive, polynomial-time framework by showing global lca-DAGs coincide with holju graphs, obtainable from a single vertex via restricted leaf-attachment rules; it also reframes the problem through leaf-extended DAGs and reveals that salient set systems (descendant, ancestor, and intermediary sets) are closed precisely for global lca-networks. The paper further connects LCA properties to the Hasse-diagram of associated clustering/descendant systems, clarifying when $G$ can be reconstructed from such data and when cluster data uniquely identify the underlying DAG (regularity). Collectively, these results advance understanding of hierarchical DAGs with globally well-defined ancestral relationships and provide practical avenues for recognition and reconstruction in applications like phylogenetics and causal networks.

Abstract

Directed acyclic graphs (DAGs) are fundamental structures used across many scientific fields. A key concept in DAGs is the least common ancestor (LCA), which plays a crucial role in understanding hierarchical relationships. Surprisingly little attention has been given to DAGs that admit a unique LCA for every subset of their vertices. Here, we characterize such global lca-DAGs and provide multiple structural and combinatorial characterizations. We show that global lca-DAGs have a close connection to join semi-lattices and establish a connection to forbidden topological minors. In addition, we introduce a constructive approach to generating global lca-DAGs and demonstrate that they can be recognized in polynomial time. We investigate their relationship to clustering systems and other set systems derived from the underlying DAGs.

Global Least Common Ancestor (LCA) Networks

TL;DR

This work characterizes global LCA-DAGs as exactly the networks whose reachability poset forms a join-semilattice, linking LCA uniqueness to semilattice structure and to topological-minor conditions. It provides a constructive, polynomial-time framework by showing global lca-DAGs coincide with holju graphs, obtainable from a single vertex via restricted leaf-attachment rules; it also reframes the problem through leaf-extended DAGs and reveals that salient set systems (descendant, ancestor, and intermediary sets) are closed precisely for global lca-networks. The paper further connects LCA properties to the Hasse-diagram of associated clustering/descendant systems, clarifying when can be reconstructed from such data and when cluster data uniquely identify the underlying DAG (regularity). Collectively, these results advance understanding of hierarchical DAGs with globally well-defined ancestral relationships and provide practical avenues for recognition and reconstruction in applications like phylogenetics and causal networks.

Abstract

Directed acyclic graphs (DAGs) are fundamental structures used across many scientific fields. A key concept in DAGs is the least common ancestor (LCA), which plays a crucial role in understanding hierarchical relationships. Surprisingly little attention has been given to DAGs that admit a unique LCA for every subset of their vertices. Here, we characterize such global lca-DAGs and provide multiple structural and combinatorial characterizations. We show that global lca-DAGs have a close connection to join semi-lattices and establish a connection to forbidden topological minors. In addition, we introduce a constructive approach to generating global lca-DAGs and demonstrate that they can be recognized in polynomial time. We investigate their relationship to clustering systems and other set systems derived from the underlying DAGs.

Paper Structure

This paper contains 13 sections, 34 theorems, 9 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Sets
  4. DAGs
  5. Clustering and Descendant Systems
  6. Least Common Ancestors (LCAs)
  7. Basic properties of global lca-networks and a first characterization
  8. A characterization by $K_{2,2}$-minors
  9. A constructive characterization
  10. A characterization by tree-leaf-child and leaf-extended DAGs
  11. Characterization via descendants, ancestors, and intermediaries
  12. The Canonical Hasse-diagram of a DAG
  13. Concluding Remarks

Key Result

Lemma 2.2

Let $G = (V,E)$ be a DAG. Then, for all $u,v\in V$, it holds that $u\prec_G v$ if and only if $u\prec_{\sf(G)} v$ and, for all $v\in V$, it holds that $\mathop{\mathrm{\mathtt{C}}}\nolimits_G(v)= \mathop{\mathrm{\mathtt{C}}}\nolimits_{\sf(G)}(v)$. In particular, $\sf(G)$ is uniquely determined for a

Figures (4)

  • Figure 1: The clustering system $\mathfrak{C}_G = \{\{x\},\{y\},\{x,y\}\}$, indicated in blue, of $G$ is closed but $G$ does not have the lca-property as $|\operatorname{LCA}_G(\{x,y\})|>1$. Note that $G$ contains a $K_{2,2}$-minor without $X$- or $X'$-subdivision, see Section \ref{['sec:Ktt']} for more details. Although $\mathfrak{C}_G$ is closed, the descendant system $\mathfrak{D}_G$ of $G$ is not closed because $\mathtt{D}_{G}(b) \cap \mathtt{D}_{G}(c) = \{b,x,y\} \cap \{c,x,y\} = \{x,y\}$ but $\{x,y\} \notin \mathfrak{D}_G$.
  • Figure 2: The three DAGs $K_{2,2}$, $X$ and $X'$ that appear in the characterization of global lca-networks in terms of minors (Thm. \ref{['thm:globalca<=>findXminor-in-K22minors']}). In addition, two global lca-networks $N$ and $N'$ are shown. Here, $N$ contains a $K_{2,2}$-minor but not a (strict $K_{2,2}$)-minor and $N'$ contains a (strict $K_{2,2}$)-minor (the respective $K_{2,2}$-minors are located within the gray-shaded area).
  • Figure 3: Three networks $N$, $N'$ and $N"$, where $N'$ is obtained from $N$ by applying \ref{['rule:add-x-N']} but not \ref{['rule:addLeaf+']}, and $N"$ is obtained from $N$ by applying \ref{['rule:addLeaf+']}. Both $N$ and $N"$ have the global lca-property, while $N'$ does not. See the text for more details.
  • Figure 4: Two networks $N$ and $N'$, neither of which have the global (pairwise)-lca-property. Moreover, $N$ does have the pairwise-lca-property i.e. $\operatorname{lca}_N(A)$ is well-defined for each subset $A\subseteq L(N)$ of size $|A|=2$, yet $\operatorname{LCA}_N(\{a,b,c\})=\{w,w'\}$ and $N$ therefore does not have the lca-property. In $N'$, there is a unique LCA for every non-empty subset of leaves, but $\operatorname{LCA}_{N'}(\{u,v\})=\{w,w'\}$, implying that $N'$ is not a global lca-network. Moreover, both $N$ and $N'$ are networks in which every inner vertex is adjacent to a leaf, indicating the need to enforce leaf-tree-child in Proposition \ref{['prop:TLC-lca-char']}. Lastly, for $N$, the set system $\mathfrak{C}_N$ is a pre-binary clustering system that is not closed since $\mathop{\mathrm{\mathtt{C}}}\nolimits_N(w)\cap \mathop{\mathrm{\mathtt{C}}}\nolimits_N(w') = \{a,b,c\} \notin \mathfrak{C}_N$. The latter argument shows that the Statements (5) and (6) in Proposition \ref{['prop:TLC-lca-char']} are, in the general setting, not equivalent.

Theorems & Definitions (74)

  • Definition 2.1
  • Lemma 2.2: HL:24
  • Lemma 2.4: S+24
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.8
  • proof
  • Definition 2.9
  • Lemma 2.10
  • ...and 64 more