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We call a point set in a complex K a 0-cell if it contains just one point of K, and a 1-cell if it is an open arc. A set L of 0-cells and 1-cells of K is called a linear graph on K if (i) no two members of L intersect, (ii) the union of all the members of L is K, (iii) each end-point of a 1-cell of L is a 0-cell of L and (iv) the number of 0-cells and 1-cells of L is finite and not 0.
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W. T. Tutte (1947) studied this question.
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