Path-Based Pendant Constructions

• The paper introduces path-based pendant constructions by systematically attaching pendant vertices to base graphs, with implications across graph theory, combinatorics, and applied physics.
• Key findings include explicit EKR thresholds and domination regimes, established through combinatorial, enumerative, and algorithmic methods.
• Methodologies extend into physical and network contexts, offering design rules for programmable droplet guidance and fault-tolerant pendant Steiner tree constructions.

Path-based pendant constructions encompass a spectrum of mathematical, combinatorial, and physical models in which additional structure—typically in the form of pendant (leaf) vertices, edges, or path-embedded motifs—is attached to base objects such as paths, trees, or surfaces. These constructions have broad applications, ranging from extremal set theory and domination in graphs, to the engineering of pinning landscapes for pendant droplets, to structural connectivity measures in network theory. The term integrates foundational ideas in graph theory, combinatorics, and applied physics, and distinct subfields focus on their combinatorial, enumerative, or physical dynamics across discrete or continuous settings.

1. Graph-Theoretic Definitions and Core Path-Based Pendant Models

In combinatorics and graph theory, a pendant construction refers to the systematic attachment of pendant (degree-one) vertices to every vertex of a given base graph, often a path or tree. For the $n$-vertex path $P_n$ with base vertex set $\{x_1,\dots,x_n\}$, the prototypical pendant path $P_n^*$ is constructed by adjoining a pendant $p_i$ to each $x_i$, yielding

$$V(P_n^*) = \{x_1, \dots, x_n\} \cup \{p_1, \dots, p_n\},\quad E(P_n^*) = \{x_ix_{i+1}\mid 1\leq i<n\} \cup \{x_ip_i\mid 1\leq i\leq n\}$$

This is the uniform pendant path (with one pendant per base), but various generalized pendant models allow the attachment of multiple leaves per base vertex or pendants following prescribed patterns (e.g., only at even or interior positions) (Carrion et al., 25 Oct 2025, Allagan et al., 7 Jan 2026).

In more advanced network-theoretic settings, pendant trees refer to Steiner trees connecting given terminal sets where each terminal is forced to be a leaf in the tree. In product networks, path-based pendant constructions underpin lower bounds for pedant tree-connectivity and the construction, by cross-fiber "gluing," of internally disjoint pendant Steiner trees spanning multiple factors (Mao, 2015).

2. EKR-Type Results and Independent Set Thresholds for Pendant Paths

A central direction in the combinatorics of path-based pendant constructions concerns Erdős–Ko–Rado (EKR)-type theorems. Given a graph $G$, EKR-type questions seek to characterize for which $r$ all largest intersecting families of independent $r$-vertex sets are "stars" (families containing all independent $r$-sets passing through a fixed vertex). For the pendant path $P_n^*$, the independence number satisfies $\alpha(P_n^*) = n$, since all pendants $\{p_i\}$ form a maximum independent set.

The threshold for $r$-EKR behavior in $P_n^*$ aligns with the half-independence principle of Holroyd–Talbot: $P_n^*$ is $r$-EKR for $1\leq r\leq \lfloor n/2\rfloor$, and not $r$-EKR for larger $r$. Explicit counterexamples for the non-EKR regime are constructed by augmenting the unique largest star (at $p_2$ or $p_{n-1}$) with an extra independent set intersecting all sets in the star but lying outside it, thus producing a strictly larger intersecting family (Carrion et al., 25 Oct 2025). The failure of the EKR property for $r>n/2$ is algorithmically transparent via direct construction and persists in extensions to generalized pendant graphs.

The table below summarizes EKR thresholds for the pendant path $P_n^*$:

Regime	$r$	EKR Property
"Safe"	$1\leq r\leq n/2$	$r$-EKR holds
"Critical"	$r>n/2$	$r$-EKR fails

The sharpness and universality of the $r\leq n/2$ threshold suggest further inquiries for cycles, path powers, and other pendant constructions.

3. Domination, Minimum Dominating Sets, and Enumeration Regimes

In domination theory, path-based pendant constructions are instrumental in controlling the number and structure of minimum dominating sets. Let $G(n, r)$ denote the path $P_n$ with $r$ pendants per base vertex; its domination number $\gamma(G(n,r))=n$ for any $r\geq1$ (one vertex per cluster enforces coverage). The dominion $\zeta(G(n,r))$—the count of all minimum dominating sets—exhibits a sharp dichotomy (Allagan et al., 7 Jan 2026):

• For $r=1$, each cluster allows a free choice (host $v_i$ or pendant $\ell_i$), yielding $\zeta=2^n=2^{\gamma}$.
• For $r\geq2$, minimality constraints force the inclusion of all hosts, so $\zeta=1$.

This sharp forcing threshold demarcates maximal flexibility and rigidity. Intermediate patterns, such as removal of endpoint pendants or alternating pendant placement, interpolate between these extremes, leading to dominion enumerations governed by cluster decompositions or Fibonacci recurrences. For example, an even-pendant path $E_n$ with pendants at even-indexed vertices satisfies $\zeta(E_n) = F_{k+1}$, where $k=\lfloor n/2\rfloor$ and $F_t$ is the $t$-th Fibonacci number.

The following summarizes key enumeration results for path-based pendant trees:

Pendant Pattern	$\gamma$	$\zeta$	Asymptotics
Uniform $r=1$	$n$	$2^n$	Exponential
Uniform $r\geq2$	$n$	$1$	Rigid
Endpoints excluded	$n-2$	$2^{n-4}=2^{\gamma-2}$	Exponential, slightly reduced
Alternating even	$k$	$F_{k+1}$	$\sim \varphi^\gamma$, Fibonacci

Algorithmic evaluation of $\gamma$ and $\zeta$ remains efficient ($O(n)$), and extension to trees or bounded-treewidth graphs is tractable.

4. Path-Based Pendant Constructions in Physical and Engineering Contexts

Beyond combinatorics, pendant constructions appear in physical experiments on pendant drops sliding along wet, tilted substrates. Here, path-based topographic defects—"tracks" or "ridges"—can steer the motion of pendant drops via gravito-capillary pinning (Jambon-Puillet, 2024). In a typical setup, a substrate is engineered with prescribed topography $s(x,y)$, spin-coated, and inverted; a drop is placed underneath and guided by defect-induced forces.

The key physical model combines gravity and capillarity with the pinning energy given by

$$E(x_d, y_d) = E_0 + \rho g\,A\,h_s\,\ell_c \left[ (\hat h_d * \hat s)(\hat x_d, \hat y_d) + (\nabla\hat h_d * \nabla\hat s)(\hat x_d, \hat y_d) \right]$$

where $A$ is drop amplitude, $h_s$ is defect height, $\ell_c$ is the capillary length, and " * " denotes 2D convolution. The pinning force $\mathbf{F}_p$ is the negative gradient of this energy, and the depinning criterion is matched when the driving force from tilt equates the maximum pinning force. By adjusting track height, width, and sharpness, complex path-following and droplet routing are programmably realized.

This framework yields closed-form design rules for programmable droplet logic and microfluidics, generalizing to arbitrary planar paths.

5. Pendant Steiner Trees and Path-Based Connectivity in Networks

In network science and reliability, the path-based methodology extends to the construction and packing of pendant Steiner trees—trees connecting a fixed terminal set $S$ where each terminal is a leaf. For two connected graphs $G$ and $H$, their Cartesian product supports the pairing of pendant Steiner tree packings in each factor ("leaf-path forests"), where three internally disjoint pendant $S$-Steiner trees can be constructed for any $|S|=3$ in $G \Box H$ via a cross-product of leaf-paths (Mao, 2015).

The pendant tree-3-connectivity $\tau_3(G)$ is the minimum number over all $S$ of size 3 of internally disjoint pendant $S$-Steiner trees in $G$. The main result establishes the tight lower bound

$$\tau_3(G\Box H) \geq \min\{3\lfloor \tau_3(G)/2 \rfloor,\ 3\lfloor \tau_3(H)/2 \rfloor\}$$

The path-based construction proceeds by factorizing both $G$ and $H$ into appropriate pendant forests, pairing, and then "gluing" along product fibers to realize the maximal number of internally disjoint configurations, providing guarantees for fault-tolerant broadcast/multicast in network topologies.

6. Threshold Phenomena, Regimes, and Open Directions

Path-based pendant constructions exhibit sharp thresholds in combinatorial, enumerative, and physical behaviors:

• EKR threshold: For independence, EKR-type properties hold up to $r \leq \lfloor n/2\rfloor$ and fail above (Carrion et al., 25 Oct 2025).
• Domination threshold: One pendant per host yields exponential dominion, two or more force rigidity (Allagan et al., 7 Jan 2026).
• Physical pinning threshold: Track heights $h_s$ must be chosen to exceed the scaled tilt parameter for robust droplet guidance (Jambon-Puillet, 2024).

Open directions include characterizing periodic or sparse pendant patterns yielding subexponential (e.g., Pisano-type) dominion growth, locating analogs of EKR/dominion thresholds in other pendant hosts (cycles, powers), and extending path-based pendant design to arbitrary networks and programmable physical media.

A plausible implication is that half-independence and local forcing principles underlie many of these thresholds, providing a unifying perspective for further study of constraint propagation and flexibility in both discrete and continuous pendant-augmented systems.