Activity-on-Vertex (AOV) Graphs

• AOV graphs are a combinatorial framework that studies graphs by analyzing vertex activities in maximal independent sets.
• They define internal and external vertex activities using substitution sets and level labellings to generate intervals in the Boolean lattice.
• This framework enables interval covers or partitions in various graph classes, offering new insights into vertex-based enumeration and graph invariants.

An Activity-on-Vertex (AOV) graph is a combinatorial framework in which the structure and properties of graphs are studied via the activities of their vertices, rather than edges. Central to this approach is the formalization of vertex activities—internal and external—which are defined relative to maximal independent sets. Through the analysis of these activities, it is possible to generate interval covers or, under suitable conditions, partitions of the Boolean lattice $2^V$, where $V$ is the vertex set of a graph. Unlike edge-based frameworks, the collection of independent sets of vertices does not form a matroid, necessitating new methods for decomposing $2^V$. The AOV graph framework extends and adapts concepts from matroid and lattice theory, most notably those inspired by the work of Crapo and Tutte, to vertex configurations.

1. Definition of Vertex Activities

Let $G = (V, E)$ be a finite simple graph, and let $A \subseteq V$ be a maximal independent set. For each $v \in A$, define the substitution set: $$Subs(v) = \{ u \in N(v) \mid (A \setminus \{v\}) \cup \{u\} \text{ is independent} \}$$ where $N(v)$ is the open neighborhood of $v$.

A vertex $v \in A$ is internally active with respect to $A$ if either:

• $Subs(v) = \emptyset$, or
• $Subs(v) \neq \emptyset$ and $v > \max\{u \mid u \in Subs(v)\}$ (with respect to a fixed labelling/order on $V$).

A vertex $v \in V \setminus A$ is externally active with respect to $A$ if there exists $a \in A$ with $v \in N(a)$ and $v > a$. The set of externally active vertices is denoted

$$Ext(A) = \{ v \in V \setminus A \mid \exists a \in A: v \in N(a),\ v > a \}$$

The sets of internally ($Int(A)$) and externally ($Ext(A)$) active vertices precisely determine the interval generated by $A$ in the Boolean lattice.

2. Interval Generation in the Boolean Lattice

Given a maximal independent set $A$, the interval generated by $A$ in $2^V$ is: $[A \setminus Int(A);\ A \cup Ext(A)]$ which is the set of all subsets $X$ such that

$$A \setminus Int(A) \subseteq X \subseteq A \cup Ext(A)$$

It is shown that the collection of such intervals, taken over all maximal independent sets $\mathcal{M}(G)$, constitutes a cover of $2^V$: $$2^V = \bigcup_{A \in \mathcal{M}(G)} [A \setminus Int(A);\, A \cup Ext(A)]$$ This result holds for any labelling, but the intervals may overlap. With appropriate labelling schemes, the intervals become disjoint, yielding a partition of the Boolean lattice.

3. Properties of Maximal Independent Sets via Activities

A maximal independent set $A$ can be characterized further in terms of its active vertices:

• Internally complete: $Int(A) = A$.
• Externally complete: $Ext(A) = V \setminus A$.
• Complete: both internally and externally complete.

If a maximal independent set $A$ is complete, then its interval spans all of $2^V$: $$[A \setminus Int(A);\, A \cup Ext(A)] = [\emptyset; V]$$ If two distinct internally complete maximal independent sets exist, their intervals overlap, and thus the generated cover cannot be a partition. These findings highlight the dependency of interval structure on the specific activity configuration of maximal independent sets.

4. Constructions in Special Graph Classes

The generation of interval covers or partitions varies across graph families:

• Complete Graphs $K_n$: Each single-vertex maximal independent set $\{i\}$, for $i=1,2,\dots, n$, generates an interval $[\{i\}; \{i, i+1, \dots, n\}]$. Appropriate labelling ensures these intervals are disjoint, effecting a partition of $2^V$.
• Joins of Complete and Empty Graphs / Threshold Graphs: Constructions for these families, often defined by specific labellings such as lexicographic or colexicographic order, are tailored so that the induced activity intervals are disjoint.

These cases exemplify how structural and labelling properties of the underlying graph influence the decomposition of $2^V$ via vertex activities.

5. Role of Level Labellings in Pruned Graphs

A pruned tree is a rooted tree where every non-leaf node has at least one leaf descendant; a pruned graph is built from such trees. A level labelling assigns labels increasing with the level: if $u$ is at a lower level than $v$, then $u < v$. Under a level labelling, only the leaves of the original tree in a maximal independent set can be internally active; non-leaf vertices are not internally active, as they can be replaced by their leaf children.

A mapping

$f(S) = S \cup [L \cap (V \setminus ch(S))]$

with $L$ the leaves and $ch(S)$ the children of $S$, establishes a bijection between independent sets missing the leaves and maximal independent sets. In this context, for a pruned tree $T$ with level labelling and pruned graph $H$, intervals constructed as $[f^{-1}(A); A \cup Ext(A)]$ (for $A \in \mathcal{M}(H)$) form a partition of $2^V$. The labelling synchronizes vertex activities across all maximal independent sets, so their generated intervals are necessarily disjoint.

6. Implications for AOV Graph Analysis and Enumeration

By extending constructions known from matroid and edge activity theory to vertices, the AOV framework provides new mechanisms for decomposing and enumerating vertex subsets in graphs. Although the family of independent sets does not comprise a matroid, the introduction of vertex activities and suitable labellings still yield interval covers or partitions of $2^V$. Every vertex subset can thus be uniquely reconstructed from an appropriately chosen maximal independent set together with its internally and externally active vertices.

This decomposition affords detailed insight into the combinatorial organization of AOV graphs, with potential generalizations for graph polynomials and invariants defined in terms of vertex activity. The interplay between activity, interval generation, and labelling forms a foundational aspect of the AOV perspective, enriching the understanding of vertex-based enumeration and structural analysis in graphs.