Edge Open Packing Number

• Edge open packing number is a graph invariant defined as the maximum set of edges forming disjoint induced stars without shared common edges.
• It employs structural decompositions and dynamic programming techniques, with analyses in trees, chordal graphs, and various product graphs.
• The concept connects to induced matching, injective edge coloring, and extremal bounds, offering insights into optimization and complexity in graph theory.

The edge open packing number of a graph $G=(V,E)$ is the maximum size of an edge set $D\subseteq E(G)$ such that no two selected edges have a common edge, that is, no third edge joins an endpoint of one selected edge to an endpoint of the other. In the prevailing notation this parameter is written $\rho_e^o(G)$, although some papers use $p_e(G)$. The defining constraint admits a structural interpretation: if $D$ is an edge open packing set, then the subgraph induced by the endpoints of the edges in $D$ is a disjoint union of induced stars. This places edge open packing between matching-type and star-packing-type invariants and explains its close relation to induced matchings, injective edge coloring, and edge-packing problems on structured graph classes (Brešar et al., 2024, Bresar et al., 2024).

1. Definitions and equivalent descriptions

Let $e_1,e_2\in E(G)$. The standard definition says that $e_1$ and $e_2$ have a common edge if there exists an edge $e\neq e_1,e_2$ such that $D\subseteq E(G)$0 joins an endpoint of $D\subseteq E(G)$1 to an endpoint of $D\subseteq E(G)$2. An edge open packing set is then a set $D\subseteq E(G)$3 in which no two edges have a common edge, and the largest possible size of such a set is $D\subseteq E(G)$4. A frequently used equivalent formulation is that for distinct $D\subseteq E(G)$5, there must be no third edge $D\subseteq E(G)$6 such that the edge-induced subgraph $D\subseteq E(G)$7 is either a triangle $D\subseteq E(G)$8 or a path $D\subseteq E(G)$9 (Santra, 17 Oct 2025).

The most useful structural reformulation is that if $\rho_e^o(G)$0 is an EOP set, then $\rho_e^o(G)$1 is a disjoint union of induced stars. This is stronger than a purely local conflict description because it identifies the exact connected components allowed inside an optimal solution. In the product-graph literature the same parameter is denoted $\rho_e^o(G)$2, but the object is the same: the maximum number of edges in an induced subgraph whose components are stars (Bresar et al., 2024).

A recurrent misconception is that edge open packing is simply the vertex open packing number of the line graph. The later product-graph study explicitly warns against that naive transfer: in $\rho_e^o(G)$3 with $\rho_e^o(G)$4, two distinct edges can have intersecting edge-neighborhoods in a line-graph sense, yet the whole edge set $\rho_e^o(G)$5 is an EOP set. This is why the induced-star characterization is the cleaner invariant-defining viewpoint (Bresar et al., 2024).

2. Relation to induced matching and edge-coloring

The closest classical invariant is the induced matching number $\rho_e^o(G)$6. An induced matching is exactly the special case in which every component of the selected induced subgraph is a copy of $\rho_e^o(G)$7. Since edge open packing allows arbitrary induced stars, every induced matching is an edge open packing, and therefore

$\rho_e^o(G)$8

The containment is strict in general: for a star $\rho_e^o(G)$9, all $p_e(G)$0 edges may belong to an EOP set, whereas an induced matching can use only one edge (Bresar et al., 2024).

This relationship is not merely definitional; it drives structural and algorithmic comparisons throughout the literature. In trees, for example, the gap between induced matching and edge open packing becomes a classification problem, while in product graphs the two parameters share analogous lower-bound constructions but differ in exact behavior because EOP solutions may retain star components of size larger than one (Bresar et al., 2024).

Edge open packing is also linked to injective edge coloring. Each color class in an injective edge coloring is an EOP set, so injective edge coloring can be viewed as a partition of $p_e(G)$1 into edge open packing sets. This places $p_e(G)$2 on the packing side of a broader decomposition theory in which one studies how conflict-defined edge families can cover all edges of a graph (Santra, 17 Oct 2025).

3. Optimization problem and complexity landscape

The associated optimization problem is the Maximum Edge Open Packing Problem: given a graph $p_e(G)$3, compute $p_e(G)$4, and possibly construct a corresponding maximum EOP set. The natural decision version asks whether $p_e(G)$5 for a prescribed integer $p_e(G)$6 (Brešar et al., 2024).

The hardness theory is already strong on restricted classes. The decision problem is NP-complete even for graphs with a universal vertex, for Eulerian bipartite graphs, and for planar graphs of maximum degree at most $p_e(G)$7. These reductions show that the parameter remains computationally difficult in dense, sparse-planar, and parity-constrained bipartite settings alike (Brešar et al., 2024). Independently, the product-graph study shows that both the induced matching problem and the edge open packing problem are NP-complete even for triangular graphs, namely graphs in which every edge lies in a triangle (Bresar et al., 2024).

At the same time, the parameter admits nontrivial fixed-parameter tractability. For chordal graphs, the 2026 algorithmic study gives an FPT algorithm parameterized by the clique number $p_e(G)$8, with running time stated in the abstract as

$p_e(G)$9

and described in the body through a $D$0 state space over clique bags. That paper also emphasizes that the complexity of MaxEOP on general chordal graphs remains open, even though several important chordal subclasses are now known to be tractable (Bhyravarapu et al., 26 Jun 2026).

4. Exact algorithms on structured graph classes

Current exact algorithms exploit the fact that EOP solutions are unions of induced stars, so decomposition structures that localize star centers and leaves become algorithmically effective.

Graph class	Complexity	Main structural device
Trees	linear time	rooted-subtree DP
Proper interval graphs	polynomial time	BCO suffix recursion
Block graphs	polynomial time	cut-tree DP
Split graphs	linear time	clique–independent-set formula
Distance-hereditary graphs	$D$1	canonical decomposition by twin sets
Biconvex bipartite graphs	$D$2	multi-chain ordering DP
Chordal graphs	FPT in $D$3	clique-bag DP

For trees, the foundational algorithm uses five rooted-subtree quantities: $D$4, $D$5, $D$6, $D$7, and $D$8, corresponding respectively to unrestricted optimum, root as star center, root as star leaf, root unused, and root together with its neighborhood untouched. The master decomposition is

$D$9

and the remaining recurrences are additive over the children of $D$0, which yields a bottom-up linear-time dynamic program (Brešar et al., 2024).

For subclasses of chordal graphs, the 2025 study answers part of an open question of Brešar and Samadi. In proper interval graphs, a bi-compatible elimination ordering allows recursive reduction to suffix subgraphs. In block graphs, a rooted cut-tree supports a bottom-up DP in which every block behaves as a clique gadget and an EOP set uses at most one edge per block. In split graphs $D$1, with $D$2 independent and $D$3 a maximum clique, the value has the exact closed form

$D$4

where

$D$5

with $D$6 and $D$7. The split-graph case is explicitly linear-time (Santra, 17 Oct 2025).

The 2026 algorithmic paper extends tractability further. For distance-hereditary graphs, the algorithm uses the canonical decomposition tree and partitions EOP sets according to how they interact with the twin set of a decomposition node, leading to an $D$8 dynamic program. For biconvex bipartite graphs, a multi-chain ordering yields a layer-by-layer DP in which no two parallel selected edges can appear between consecutive layers and the leaves of every solution star form contiguous blocks in both layer orderings; the resulting implementation is polynomial, with explicit bound $D$9. For chordal graphs, the FPT algorithm relies on the fact that a clique bag can contain at most one selected internal edge (Bhyravarapu et al., 26 Jun 2026).

5. Structural bounds and extremal characterizations

Several basic extremal facts are now standard. For connected graphs,

$e_1,e_2\in E(G)$0

There is also a general diameter lower bound,

$e_1,e_2\in E(G)$1

which reappears in later characterization theorems as a necessary condition on fixed-$e_1,e_2\in E(G)$2 instances (Pandey et al., 3 Aug 2025).

A central extremal inequality is

$e_1,e_2\in E(G)$3

Equality was characterized completely in 2024: it holds if and only if either $e_1,e_2\in E(G)$4 is a disjoint union of stars or $e_1,e_2\in E(G)$5 belongs to a specific bipartite family $e_1,e_2\in E(G)$6 in which $e_1,e_2\in E(G)$7 has bipartition $e_1,e_2\in E(G)$8, minimum degree $e_1,e_2\in E(G)$9, and every vertex of $e_1$0 has exactly one neighbor in $e_1$1 and $e_1$2 neighbors in $e_1$3 (Brešar et al., 2024).

The same paper establishes sharp edge-deletion bounds. If $e_1$4 and $e_1$5, then

$e_1$6

while for $e_1$7 one has

$e_1$8

Every admissible value in these intervals is realizable by a connected graph and a designated edge (Brešar et al., 2024).

Later work pushed the structural theory much further. For any fixed integer $e_1$9, connected graphs with $e_2$0 are characterized by four conditions: a diameter bound $e_2$1; exclusion of induced stars $e_2$2 with $e_2$3; a blocking condition on induced matchings of size $e_2$4; and an analogous blocking condition on size-$e_2$5 EOP configurations whose induced subgraph is a union of several stars. The corresponding equality criterion states that, apart from small stars, $e_2$6 precisely when the graph satisfies the $e_2$7-conditions but fails at least one of the $e_2$8-conditions (Pandey et al., 3 Aug 2025).

Near-maximum values have also been catalogued. For connected graphs with $e_2$9,

$e\neq e_1,e_2$0

if and only if $e\neq e_1,e_2$1 is obtained from $e\neq e_1,e_2$2 by subdividing exactly one edge once; $e\neq e_1,e_2$3 if and only if $e\neq e_1,e_2$4 belongs to seven families $e\neq e_1,e_2$5; and $e\neq e_1,e_2$6 if and only if $e\neq e_1,e_2$7 belongs to one of the $e\neq e_1,e_2$8 explicitly defined families

$e\neq e_1,e_2$9

These classifications make the “almost all edges can be selected” regime unusually explicit for a nontrivial edge-packing parameter (Pandey et al., 3 Aug 2025).

6. Product graphs, hypercubes, and specialized structural theory

A parallel line of work studies edge open packing together with induced matching under graph products. In that paper the parameter is denoted $D\subseteq E(G)$00. Its first major result is a tree characterization: $D\subseteq E(G)$01 where $D\subseteq E(G)$02 is a spider-based family obtained by joining subdivided stars through their centers while preserving at least two leaves in each constituent spider. This identifies precisely when allowing larger star components does not improve upon induced matching in trees (Bresar et al., 2024).

For the lexicographic product, the induced matching number has the exact formula

$D\subseteq E(G)$03

while edge open packing satisfies the sharp bounds

$D\subseteq E(G)$04

For the direct product,

$D\subseteq E(G)$05

and for the Cartesian and strong products,

$D\subseteq E(G)$06

These bounds are stated to be sharp on infinite families (Bresar et al., 2024).

The same study gives sharp two-sided bounds for rooted products,

$D\subseteq E(G)$07

while explicitly leaving the search for a closed formula for $D\subseteq E(G)$08 as an open problem. It also obtains exact hypercube values: if $D\subseteq E(G)$09 with $D\subseteq E(G)$10, then

$D\subseteq E(G)$11

This is one of the cleanest exact formulas currently known for a large nontrivial family (Bresar et al., 2024).

Taken together, these results show that the edge open packing number has developed from a local edge-conflict invariant into a broader theory with three distinct facets: a star-based structural core, a rapidly expanding exact-algorithmic territory on decomposable graph classes, and a detailed extremal and product-graph calculus. The main unresolved themes suggested by current work are the full complexity of MaxEOP on chordal graphs, sharper algorithms on wider hereditary classes, and further exact formulas outside the families already classified (Santra, 17 Oct 2025, Bhyravarapu et al., 26 Jun 2026).