On the complexity of finding a sun in a graph

The sun is the graph obtained from a cycle of length even and at least six by adding edges to make the even-indexed vertices pairwise adjacent. Suns play an important role in the study of strongly chordal graphs. A graph is chordal if it does not contain an induced cycle of length at least four. A graph is strongly chordal if it is chordal and every even cycle has a chord joining vertices whose distance on the cycle is odd. Farber proved that a graph is strongly chordal if and only if it is chordal and contains no induced suns. There are well known polynomial-time algorithms for recognizing a sun in a chordal graph. Recently, polynomial-time algorithms for finding a sun for a larger class of graphs, the so-called HHD-free graphs, have been discovered. In this paper, we prove the problem of deciding whether an arbitrary graph contains a sun in NP-complete.

💡 Research Summary

The paper investigates the computational complexity of detecting a “sun” in an arbitrary graph. A sun (more precisely a k‑sun for k ≥ 3) is obtained from an even‑length cycle of length 2k by making all even‑indexed vertices pairwise adjacent; the resulting graph consists of a clique of k “center” vertices, a stable set of k “ears”, and edges joining each ear to its two consecutive centers. Strongly chordal graphs are exactly the chordal graphs that contain no induced sun (Farber’s theorem), and they admit a simple elimination ordering.

While polynomial‑time algorithms exist for recognizing suns in chordal graphs and in the broader class of HHD‑free graphs (graphs without a house, a hole, or a domino), the complexity for unrestricted graphs remained open. The authors settle this by proving three main results.

Theorem 2 – NP‑completeness of Sun Detection.
The reduction starts from the NP‑complete problem “Stable Set in Triangle‑Free Graphs”. Given a triangle‑free graph G and an integer k, they construct a new graph f(G,k) as follows:

• Replace each original vertex vi by a clique Vi = {vi¹,…,viᵏ}.
• Add a clique W = {u₁,w₁,…,u_k,w_k} and a stable set X = {x₁,…,x_k}.
• For each i and j (1 ≤ i ≤ n, 1 ≤ j ≤ k) connect viʲ to uj and wj.
• For each i, connect xi to wi and to u_{i+1} (indices modulo k).

Two crucial observations hold: (1) no triangle in f(G,k) can have vertices from three distinct Vi’s, and (2) any two vertices from the same Vi have identical neighborhoods inside W. Using these properties, the authors prove a bidirectional correspondence: G contains a stable set of size k iff f(G,k) contains a 2k‑sun. The forward direction builds a sun by selecting one vertex from each Vi belonging to the stable set and using the corresponding xi’s as ears. The reverse direction shows that any sun in f(G,k) must have exactly k ears in the union U = ⋃Vi and the remaining k ears in X; structural constraints force the ears in U to come from distinct Vi’s, which then correspond to a stable set in G. Since the source problem is NP‑complete, sun detection is NP‑complete.

Theorem 3 – Hardness Persists Without Large Antiholes.
The authors strengthen the result by showing that f(G,k) never contains an induced antihole of length t ≥ 7. Assuming the contrary, they analyze the placement of the antihole’s vertices among U, W, and X, using the same observations about neighborhoods. A careful case analysis leads to contradictions, establishing that even in graphs excluding large antiholes the sun‑finding problem remains NP‑complete.

Theorem 4 – k‑SUN is NP‑complete.
Here the authors reduce the classic k‑CLIQUE problem to k‑SUN. Given a graph G, they construct h(G) by adding, for each edge ab, a new vertex v(a,b) of degree two adjacent to a and b. The set Y of all such degree‑two vertices forms a stable set. If G has a k‑clique, then the k clique vertices together with any k distinct vertices from Y induce a k‑sun (the clique vertices are the centers, the Y‑vertices are the ears). Conversely, any k‑sun in h(G) must use only original vertices as centers because ears have degree two, so the centers form a k‑clique in G. Since k‑CLIQUE is NP‑complete, k‑SUN is NP‑complete as well. Consequently, deciding whether an arbitrary graph contains a sun reduces to solving O(n) instances of k‑SUN for k = 3,…,⌊n/2⌋, confirming the overall NP‑completeness.

The paper concludes by situating these results within the broader landscape of graph classes. While chordal and HHD‑free graphs admit efficient sun detection, the NP‑completeness for unrestricted graphs (and even for graphs without large antiholes) delineates the limits of tractability. Moreover, the authors note that for weakly chordal and perfect graphs, polynomial‑time sun recognition remains possible, highlighting the nuanced interplay between structural graph properties and algorithmic complexity.

In summary, the work establishes that the seemingly modest task of finding a sun—a structure central to the characterization of strongly chordal graphs—is computationally intractable in general, thereby providing a clear boundary between the algorithmically manageable subclasses and the full class of graphs. This contributes a fundamental complexity classification to the theory of chordality and enriches our understanding of how specific induced subgraphs influence the difficulty of graph‑theoretic decision problems.