---
title: Path-Star Graph Problem
url: https://www.emergentmind.com/topics/path-star-graph-problem
type: topic
---

# Path-Star Graph Problem

The path-star graph problem encompasses a collection of questions concerning the structure, extremal properties, and spectral characteristics of graphs and graph families that combine path and star subgraphs. This theoretical landscape provides a unifying framework for Turán-type extremal questions, spectral extremal graph theory, and structural graph theory centered on mixtures of paths and stars, commonly referred to as "path-star forests." The problem domain includes classical edge-extremal, spectral, and order-theoretic constraints on graphs that avoid prescribed combinations of paths and stars as (not necessarily induced) subgraphs, as well as characterizing configurations that optimize or avoid these forbidden structures.

## 1. Definitions and Core Notation

A path of order $\ell$ is denoted $P_\ell$, a tree on $\ell$ vertices of maximum degree 2. A star $S_a$ is $K_{1,a}$, a tree on $a+1$ vertices with one central vertex of degree $a$ joined to $a$ leaves. A *linear forest* is any disjoint union of paths; a *star forest* is a disjoint union of stars. A *path-star forest* $F$ is defined as
\[
F = \bigcup_{i=1}^p P_{\ell_i}\;\cup\;\bigcup_{j=1}^q S_{a_j}
\]
with $\ell_1\ge \ell_2\ge\dots\ge \ell_p\ge 2$ and $a_1\ge a_2\ge\dots\ge a_q\ge 3$. The Turán number $ex(n,F)$ is the maximum number of edges in an $n$-vertex graph that is $F$-free. The spectral radius $\lambda_1(G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix. For extremal spectral questions, forbidding star-path forests determines the maximum $\lambda_1(G)$ for $n$-vertex $F$-free graphs [2302.11839, 2305.11680].

## 2. Turán Numbers for Path-Star Forests

For sufficiently large $n$, the Turán number $ex(n,F)$ for an arbitrary path-star forest $F$ is characterized by a dichotomy of regimes:

- **Regime I ("paths $+$ stars"):** If
  \[
  \beta_F = q + \delta_F - \mu_F > \max_j f_j, \quad\text{where}\quad \delta_F = \sum_{i=1}^p \left\lfloor \frac{\ell_i}{2} \right\rfloor,\; \mu_F = 
    \begin{cases}
      1 & p=1,\;\ell_1\text{ even or }p\ge2\;\&\;\exists \ell_i\ne3 \\
      \frac{1}{2} & \text{otherwise}
    \end{cases},\quad f_j = (j-1) + \frac{a_j-1}{2}
  \]
  then
  \[
  ex(n,F) = \binom{q}{2} + q(n-q) + ex(n-q,\cup_i P_{\ell_i})
  \]

- **Regime II ("pure stars"):** If $\beta_F \le \max_j f_j$, then
  \[
  ex(n,F) = \max_{1\le j\le q} \left\{ (j-1)\left(n-\frac{j}{2}\right) + \left\lfloor \frac{a_j-1}{2}(n-j+1) \right\rfloor \right\}
  \]
  These cases generalize and recover the classical Turán formulas for linear forests and star forests [2305.11680].

### Extremal Graph Constructions

- In Regime I, the unique extremal graph is $K_q \vee H$, with $H$ any extremal graph for the corresponding pure-path-forest on $n-q$ vertices.
- In Regime II, the extremal graph is $K_{j^*-1} \vee (n-j^*+1,S_{a_{j^*}})$ for the maximizing $j^*$.

## 3. Spectral Extremal Results for Forbidden Path-Star Forests

Given $F$ a (path, star, or path-star) forest, let $\mathrm{Ex}_{\mathrm{sp}}(n,F)$ denote the family of $n$-vertex $F$-free graphs maximizing the spectral radius. For several key classes, the maximal $\lambda_1(G)$ is exactly achieved by the same extremal graphs as in the Turán (edge-count) sense:

- For
  \[
  F = k S_{\ell-1} \cup P_\ell
  \]
  and $n$ sufficiently large, the unique extremal graphs are the join $S_{n,h} = K_h \vee \overline{K}_{n-h}$ with $h = k + (\ell-1) - 1$, or $S^+_{n,h}$ (one edge added in the independent part) for odd $\ell$.
- For $k_1 S_{2\ell-1} \cup k_2 P_{2\ell}$ and $kS_4 \cup 2 P_5$, one gets $K_{p} \vee \overline{K}_{n-p}$ or $K_p \vee \overline{K}_{n-p}$ plus a single extra edge as the unique extremal constructions.

The proofs use Rayleigh-quotient arguments on the Perron vector, eigenvalue-interlacing, the Hong–Shu–Fang bound, and rely crucially on structural features of join graphs [2302.11839].

## 4. Ore-Type Path-Star Theorems (Hamiltonian Paths vs. Induced Stars)

The path-star graph problem in the context of Hamiltonicity is encapsulated by sharp Ore-type results: For any $t \ge 5$, a connected graph $G$ of order $n$ with $\sigma_2(G) > \frac{t-3}{t-2}n$ (where $\sigma_2(G)$ is the minimum degree-sum of two nonadjacent vertices) must have *either* a Hamiltonian path *or* an induced copy of $K_{1,t}$. This threshold is best possible and is witnessed by the bipartite extremal graphs $H_t = K_{t-3, t-1}$. At equality, the extremal graphs are joins $H \vee \overline{K}_{t-1}$, and the structure theorem describes the precise forbidden configurations [2001.00385].

## 5. Structural and Recognition Aspects

The study of star-path and path-star configurations extends beyond extremal and spectral theory to algorithmic graph theory. The characterization of star pairwise compatibility graphs (star-PCGs) provides an explicit structural recognition scheme:

- A graph $G$ admits a "star-PCG" representation (there exists a weighting of a star tree and distance thresholds giving $G$ as the threshold graph) if and only if $G$ admits a linear, "gap-free" vertex ordering. Efficient ($O(n^6)$ worst-case) recognition is possible through consecutive and contiguous orderings of relevant set systems, PQ-tree frameworks, and interval-core decompositions [1804.02895].

## 6. Broader Implications and Open Problems

The path-star graph problem reveals a robust structural dichotomy in extremal graph theory: for broad combinations of paths and stars, both the edge-maximal and spectral-maximal graphs are precisely joins of cliques with independent sets, possibly plus local corrections. This indicates a deep interplay between the presence of long paths and high-degree vertices (star centers), yielding universal constructions for forbidden path-star subgraphs.

Many open problems remain:
- Determining exact thresholds and uniqueness criteria for spectral extremal graphs as $n$ varies and as more general path-star forests are forbidden [2302.11839].
- Developing comprehensive "spectral-Turán" dictionaries for all acyclic graphs [2302.11839].
- Extending Turán and Ore-type theory to induced-subgraph settings (e.g., for arbitrary induced star-path mixtures) and to directed or weighted settings.

## 7. Representative Examples and Special Cases

| Path-Star Structure                  | Extremal Graph Construction           | Reference                        |
|--------------------------------------|--------------------------------------|----------------------------------|
| $P_\ell$ (single long path)          | Union of $(\ell-1)$-cliques          | [2305.11680]                     |
| Star forest $\cup_j S_{a_j}$         | $K_{j^*-1} \vee (n-j^*+1, S_{a_{j^*}})$ | [2305.11680], [2302.11839]       |
| $\cup_{i=1}^p P_{\ell_i} \cup \cup_{j=1}^q S_{a_j}$ | $K_q \vee H_{\mathrm{lin}}$ (see main text) | [2305.11680]             |
| $k S_{\ell-1} \cup P_\ell$           | $K_h \vee \overline{K}_{n-h}$        | [2302.11839]                     |
| $\sigma_2(G) > \frac{t-3}{t-2}n$ (Hamiltonicity criteria) | $K_{t-3, t-1}$, $H \vee \overline{K}_{t-1}$ | [2001.00385]           |

The landscape of the path-star graph problem synthesizes edge, spectral, and order-theoretic perspectives, offering a blueprint for the interaction of long "linear" and "hubbed" substructures in extremal graph theory.

Source: https://www.emergentmind.com/topics/path-star-graph-problem