On The Sitting Closer to Friends than Enemies Problem in Trees and An Intersection Model for Strongly Chordal Graphs
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A signed graph is a graph with a sign assignment to its edges. The Sitting Closer to Friends than Enemies (SCFE) problem is to find an injection of the vertex set of a given signed graph into a metric space such that for every pair of incident edges with different signs the end vertices of the positive edge are injected closer in the space than the end vertices of the negative edge. Such an injection is called a valid distance drawing. In this document, we study the SCFE problem in real trees (also known as R-trees). We show that a complete signed graph has a valid distance drawing in a real tree if and only if its subgraph composed of all (and only) its positive edges is strongly chordal. Furthermore, as an instrumental result, we show that the set of strongly chordal graphs is equal to the set of graphs with an intersection model of unit balls, proper balls, and balls in a real tree.
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