I am trying to find an example for the Theorem 5.1 of the paper "Combinatorial Local Planarity and the Width of Graph Embeddings" that can be found at http://www.fmf.uni-lj.si/~mohar/Reprints/1992/BM91_CJM44_Mohar_GraphEmbeddings.pdf. Given G(V,E) minimally embeddable on an orientable surface $\Sigma_g$ of genus $g \geq 2$. Suppose that $C_1,C_2,\ldots, C_k$ be a planarly nested sequence (defined below) and $k\leq g$. I am trying to construct an example for graphs of genus $g \geq 2$ and $ k = g$ in which none of $C_1,C_2,\ldots, C_k$ bounds disks.
Definition: A sequence of disjoint cycles $C_1,C_2, \ldots, C_k$ is planarly nested if each cycle $C_i (1 \leq i \leq k)$ has a relative $C_i$ component $H_i$ such that $H_1 \supset H_2 \supset \ldots \supset H_k$ and the graph obtained from $G$ by contracting to a single vertex all edges in the relative component $H_k$, except it feet, is planar.
A relative $C$-component for a cycle $C$ is an edge $e \in E(G)\backslash E(C)$ with both endpoint in $C$, or a connected component of $V(G) - V(C)$ together with all edges between this component and $C$. Edge of a $C$-component $R$ having an endpoint in $C$ is a foot of $R$.
Here is an example of two relatives $C$-component $H_1$ and $H_2$ of G.