Square-Free 1-11-Representations

• Square-free 1-11 representations are graph encodings that use words over vertex alphabets, avoiding consecutive repeated blocks while enforcing run-length constraints.
• They reveal structural limitations, as some graphs necessarily include squares in minimal representations despite achieving cube-freeness in permutational forms.
• The study bridges combinatorics on words, automata theory, and graph representation, offering insights into pattern avoidance in practical graph encodings.

Square-free 1-11-representations refer to representations of graphs via words over their vertex alphabets such that the adjacency structure is encoded by specific run-length constraints (the 1-11 pattern), and so that the representing word is square-free, i.e., contains no repeated consecutive nonempty blocks. This concept sits at a deep crossroads of combinatorics on words, automata theory, and graph representation theory, combining pattern avoidance constraints with graph-encoding languages.

1. Fundamental Definitions and Notation

A finite, simple, undirected graph $G=(V,E)$ is said to have a 1-11-representation if there exists a word $w$ over the alphabet $V$ such that for all distinct $x,y \in V$, the following holds:

• Let $w_{\{x,y\}}$ denote the subword of $w$ formed by deleting all letters except $x$ and $y$.
• The adjacency relation is determined by the maximum run-length of equal adjacent letters in $w_{\{x,y\}}$:

$$x \sim y \ \text{if and only if} \ \operatorname{occ}_{xx}(w_{\{x,y\}}) + \operatorname{occ}_{yy}(w_{\{x,y\}}) \le 1$$

Here, $\operatorname{occ}_{zz}(w)$ counts the (possibly overlapping) occurrences of $zz$ in $w$ for $z \in \{x,y\}$.

A word $w$ over $V$ is square-free if it contains no factor $UU$ for any nonempty word $U$.

A square-free 1-11-representation of $G$ consists of a word representing $G$ in the above sense and being square-free.

2. Structural Properties and Characterizations

Every graph admits at least one 1-11-representation, and the set of all such representations forms a regular language (Das et al., 28 Jan 2026). Square-freeness, however, is a strong restriction on the possible representatives:

• For some graphs, there exists a square-free 1-11-representation (e.g., $K_2$ with the word $12$).
• For others, particularly certain multipartite or disconnected graphs, all minimal (in length or number of concatenated permutations) 1-11-representations necessarily contain squares or even cubes.

The minimal representative is captured by the 1-11-representation number $\mathcal{R}(G)$, defined as the shortest possible length of a word $w$ that serves as a 1-11-representation of $G$.

3. Cube and Square Patterns in 1-11-Representations

A central topic concerns the degree to which repetition patterns (specifically squares and cubes) are avoidable:

• In the setting of permutational representations—words formed as a concatenation of $k$ permutations of the vertex set—the following holds (Das et al., 28 Jan 2026):
  • Any cube (factor $XXX$) in such a word, where $|X|$ is not a multiple of $|V|$, can be removed without changing the represented graph. As a result, any permutational 1-11-representation attains its minimal $k$ while being cube-free.
  • This property fails for squares: certain graphs (e.g., the disjoint union of $K_3$ and a single vertex, $G = K_3 \cup \{v\}$) are such that every minimal permutational 1-11-representation must contain a square.

This demonstrates a structural dichotomy: cube-freeness in permutational 1-11-representations can always be imposed but square-freeness is not always attainable at minimal length or permutation count.

4. Characterization Results and Examples

Key classification results and illustrative examples establish the landscape:

• For the graph $K_2$ ($V=\{1,2\}, E=\{\{1,2\}\}$), $w=12$ is both a square-free word and a 1-11-representation.
• For the graph $G=K_3 \cup \{v\}$ ($V=\{1,2,3,v\}$), every minimal-length 1-11-representation (length 6) necessarily contains a square, and every minimal permutational 1-11-representation (three permutations) also contains a square (Das et al., 28 Jan 2026).

A summary table is provided:

Graph Type	Square-free 1-11-Representation Exists?	Minimal Permutational Square-free Exists?
$K_2$	Yes	Yes
$K_3 \cup \{v\}$	No	No

5. Regular Language Structure and Automata-Theoretic Perspective

The set of all 1-11-representations for a fixed graph $G$ forms a regular language:

• For each adjacent pair $\{a,b\}$, enforce the acceptance condition that $w_{\{a,b\}}$ contains at most one occurrence of $aa$ or $bb$.
• For every non-adjacent pair, ensure at least two occurrences among $aa$ and $bb$ (or one of each).
• Intersection and union of these conditions over all pairs yields a regular language, as regular languages are closed under finite intersection.

Restricting to the sublanguage of permutational 1-11-representations (concatenations of permutations), one still obtains a regular language.

This automata-theoretic viewpoint facilitates algorithmic generation and recognition of such representations.

6. Connections to p-Complete Square-free Word-Representations and Generalizations

The paradigm of avoiding squares in restricted subwords appears in other domains of graph-encoding via words, notably in the setting of $p$-complete square-free word-representable graphs (Das et al., 8 May 2025). There, the strongest form ($p=1$) requires that no subword over any subset of the alphabet contains a square of length at least $1$, reducing to the setting where the only admissible graphs are complete graphs.

• In contrast, the 1-11 condition is significantly weaker and allows all graphs to be represented, albeit often at the cost of requiring squares in the representing word.

This demonstrates that imposing banishment of all squares in subwords massively restricts the class of representable graphs, sometimes to the trivial class.

7. Combinatorial and Pattern Avoidance Implications

Square-free 1-11-representations are of combinatorial interest as they illuminate the trade-offs between pattern-avoidance and information encoding capacity. The inability to always attain square-freeness in minimal representations signifies an intrinsic limitation in encoding certain adjacency constraints tightly without incurring repetitions. Furthermore, the language-theoretic regularity of all such representation sets establishes a firm foundation for further automata-theoretic and combinatorial-on-words analysis.

The study of 1-11-representations, and particularly their square-free instantiations, thus provides a nuanced template for exploring the interface between discrete mathematics, pattern avoidance theory, and formal language theory (Das et al., 28 Jan 2026, Das et al., 8 May 2025).