On Patterns and Languages in 1-11-Representations of Graphs

A 1-11-representation of a graph $G(V,E)$ is a word over the alphabet $V$ such that two distinct vertices $x$ and $y$ are adjacent if and only if the restricted word $w{x,y}$ (obtained from $w$ by deleting all letters except $x$ and $y$) contains at most one occurrence of $xx$ or $yy$. Although every graph admits a 1-11-representation, the repetition patterns that may or must appear in such representations have not been fully studied. In this paper, we study cube-free and square-free 1-11-representations of graphs. We first show that cubes cannot always be avoided in 1-11-representations of minimum length by providing a graph for which every minimum-length 1-11-representation necessarily contains a cube. We then focus on permutational 1-11-representations, where the representing word is a concatenation of permutations of the vertex set. In this setting, we prove that any cube appearing in a permutational 1-11-representation can be removed without changing the represented graph. As a consequence, every permutational 1-11-representation attaining the permutational 1-11-representation number is cube-free. We further show that this behaviour does not extend to squares by providing a graph for which every permutational 1-11-representation with the minimum number of permutations necessarily contains a square. Finally, we prove that the language of all 1-11-representations of a given graph is regular. Moreover, we show that the language of all permutational 1-11-representations of a graph is also regular.

💡 Research Summary

The paper investigates the occurrence of repetition patterns—specifically cubes (three consecutive identical letters) and squares (two consecutive identical letters)—in 1‑11 representations of graphs. A 1‑11 representation of a graph G = (V, E) is a word w over the alphabet V such that for any distinct vertices x and y, the restricted word w{x,y} contains at most one occurrence of the factor xx or yy if and only if x and y are adjacent; otherwise it must contain at least two occurrences of one of those factors. While every finite graph admits a 1‑11 representation, the authors ask whether cubes or squares can always be avoided, especially in representations of minimal length or minimal number of permutations.

The first main result shows that cubes cannot always be avoided in minimum‑length 1‑11 representations. The authors construct a tiny disconnected graph consisting of a triangle K₃ together with an isolated vertex v. They prove that the 1‑11 representation number R(G) equals 6 and that every length‑6 representation necessarily contains the factor “vvv”, i.e., a cube. The argument relies on the fact that for each non‑adjacent pair {v, x} (x ∈ {1,2,3}) the 1‑11 condition forces at least two occurrences of xx or yy, which forces v to appear three times in any optimal word, creating an unavoidable cube.

Having established that cubes may be forced in optimal representations, the paper turns to the subclass of permutational 1‑11 representations, where the representing word is a concatenation of permutations of the whole vertex set V. Lemma 16 proves a structural restriction: if a cube X X X appears in a permutational representation of a graph on n vertices, then the length of the block X must be a multiple of n. The proof distinguishes the case |X| < n (which would leave some vertex absent from X, leading to a contradiction with the 1‑11 condition) and the case |X| = i·n + ℓ with 1 ≤ ℓ < n, showing that the distribution of vertex occurrences across the three copies of X cannot satisfy the required adjacency constraints.

Lemma 17 demonstrates that squares formed by two identical permutation blocks can be eliminated without affecting the represented graph. If a word w contains a factor P P where P is a permutation of V, deleting one copy of P yields a shorter word w′ that still 1‑11‑represents the same graph. The proof checks adjacency preservation (alternation of each vertex pair remains) and non‑adjacency preservation (the required number of xx or yy occurrences is unchanged).

From these lemmas the authors derive that any permutational 1‑11 representation achieving the permutational 1‑11 representation number Rπ(G) (the minimum number of permutations needed) is automatically cube‑free. In other words, when we minimize the number of permutation blocks, any cube that would appear must have a block length not divisible by n, which is impossible by Lemma 16; therefore optimal permutational representations contain no cubes.

In contrast, squares do not enjoy the same freedom. The authors present a specific graph (a variation of a complete graph with additional vertices) for which every permutational 1‑11 representation that uses the minimum number of permutations inevitably contains a square. This shows that minimizing the number of permutations does not guarantee square‑freeness, highlighting a fundamental asymmetry between cubes and squares in the permutational setting.

The final part of the paper addresses the language-theoretic aspect. For a fixed graph G, let L(G) be the set of all words that 1‑11‑represent G, and let Lπ(G) be the set of all permutational 1‑11 representations of G. The authors construct deterministic finite automata (DFA) that track, for each unordered pair {x, y}, the count of occurrences of xx and yy seen so far, ensuring that the count never exceeds one for adjacent pairs and never falls below two for non‑adjacent pairs. For the permutational case, the DFA also enforces that each block of length |V| is a permutation of V. Because both conditions can be expressed with a finite number of states, the languages L(G) and Lπ(G) are regular. This extends the known result that the set of word‑representations of a graph is regular to the broader class of 1‑11 representations.

Overall, the paper contributes several key insights:

1. Cube inevitability in minimal‑length 1‑11 representations (Theorem 15).
2. Structural constraints on cubes in permutational representations (Lemma 16) and a method to delete redundant permutation squares (Lemma 17).
3. Cube‑freeness of optimal permutational representations versus square‑inevitability in some optimal cases.
4. Regularity of the full set of 1‑11 representations and of permutational 1‑11 representations for any graph.

These findings deepen our understanding of how repetition patterns interact with graph encoding schemes, reveal a nuanced difference between cubes and squares, and open avenues for algorithmic applications such as graph compression, pattern‑avoidance generation, and automata‑based verification of graph encodings.