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 \mathbb{R 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 .
A chord in a cycle is an edge that is not part of the cycle but connects two vertices of the cycle. A graph ischordal if all its cycles of four or more vertices have a chord. A graph isstrongly chordal if it is chordal, and every cycle of even length larger than 6 has an odd chord (a chord that connects two vertices that are an odd distance apart in the cycle). In this work, we show that a graph is strongly chordal if and only if it has an intersection representation by unit balls in areal tree (a metric space with a tree-like structure), or equivalently, by proper balls, or arbitrary balls. These characterizations of strongly chordal graphs find their motivation in theSitting Closer to Friends than Enemies (SCFE) problem. A signed graph is a graph with a sign assignment to its edges. The 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. Using the characterization for strongly chordal graphs, 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 .
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chord
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in a cycle is an edge that is not part of the cycle but connects two vertices of the cycle. A graph is
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chordal
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if all its cycles of four or more vertices have a chord. A graph is
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strongly chordal
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if it is chordal, and every cycle of even length larger than 6 has an odd chord (a chord that connects two vertices that are an odd distance apart in the cycle). In this work, we show that a graph is strongly chordal if and only if it has an intersection representation by unit balls in a
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real tree
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(a metric space with a tree-like structure), or equivalently, by proper balls, or arbitrary balls. These characterizations of strongly chordal graphs find their motivation in the
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Sitting Closer to Friends than Enemies
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(SCFE) problem. A
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Sitting Closer to Friends than Enemies (SCFE )
SCFE
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,
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,
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valid distance drawing. In this document, we study the SCFE problem in real trees (also known as \mathbb{R
valid distance drawing
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. Using the characterization for strongly chordal graphs, we
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,
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,
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. 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