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Given positive integers \(p\) and \(q\), a \((p, q)\)-colouring of a graph \(G\) is a mapping \(\theta: V(G) \rightarrow \{1, 2, \ldots, q\}\) such that \(\theta(u) \neq \theta(v)\) for all distinct vertices \(u, v\) in \(G\) whose distance \(d(u, v) \leq p\). The \(p\)th order chromatic number \(\chi^{(p)}(G)\) of \(G\) is the minimum value of \(q\) such that \(G\) admits a \((p, q)\)-colouring. \(G\) is said to be \((p, q)\)-maximally critical if \(\chi^{(p)}(G) = q\) and \(\chi^{(p)}(G + e) > q\) for each edge \(e\) not in \(G\).
In this paper, we study the structure of \((2, q)\)-maximally critical graphs. Some necessary or sufficient conditions for a graph to be \((2, q)\)-maximally critical are obtained. Let \(G\) be a \((2, q)\)-maximally critical graph with colour classes \(V_1, V_2, \ldots, V_q\). We show that if \(|V_1| = |V_2| = \cdots = |V_k| = 1\) and \(|V_{k+1}| = \cdots = |V_q| = h \geq 1\) for some \(k\), where \(1 \leq k \leq q-1\), then \(h \leq h^*\), where
\[h^* = \max \left\{k, \min\{q – 1, 2(q – 1 – k)\}\right\}.\]
Furthermore, for each \(h\) with \(1 \leq h \leq h^*\), we are able to construct a \((2, q)\)-maximally critical connected graph with colour classes \(V_1, V_2, \ldots, V_q\) such that \(|V_1| = |V_2| = \cdots = |V_k| = 1\) and \(|V_{k+1}| = \cdots = |V_q| = h\).