For every odd integer m >= 5, let G_m be the Mycielski graph of the cycle C_m with an additional edge joining the shadow copies of two adjacent cycle vertices. We prove that G_m is connected, has exactly one triangle, and is a minimal non-word-representable graph under induced-subgraph containment, while its line graph is word-representable. In fact, G_m is 4-vertex-critical. This gives connected examples of clique number three for a follow-up question of Mozhui, Dwary and Krishna. The smallest member of the family has eleven vertices and twenty-one edges. The proof combines a restriction lemma, a one-vertex extension of their explicit line-graph orientation, and a general vertex-criticality lemma for a single shadow-edge Mycielski extension.

The older question recorded as GRAPH-049 in UnsolvedMath was already answered negatively by Mozhui, Dwary and Krishna (2025); that answer is not claimed here as new. The contribution is the connected single-triangle family. The disconnected clique-three case is immediate by taking a disjoint triangle. No minimum-order result or answer to the separate all-triangle-free-root question is claimed. This is an unrefereed, AI-assisted preprint with a written self-audit and exact reproducibility checks; no independent review, formal certification or absolute priority is claimed.

Version 1.1 adds the vertex-criticality theorem and induced-minimality corollary, with constructive deletion-coloring certificates. It is a strengthened version of the same manuscript, not a separate paper.
