Technical Report CS0854

Authors: S. Even, A. Litman and A. Rosenberg
Abstract: This paper considers two properties of graphs, one geometrical and one topological, and shows that they are strongly related. Let $G$ be a graph with four distinguished and distinct vertices, $w_1,w_2,b_1,b_2$. Consider the two properties, $TRI^+(G)$ and $MONO(G)$, defined as follows:
$TRI^+(G)$: There is a planar drawing of $G$ such that:
  • all 3-cycles of $G$ are faces;
  • all faces of $G$ are triangles except for the single face which is the 4-cycle $(w_1-b_1-w_2-b_2-w_1)$.
$MONO(G)$: $G$ contains the 4-cycle $(w_1-b_1-w_2-b_2-w_1)$, and for any labeling of the vertices of $G$ by the colors {white, black}, such that $w_1$ and $w_2$ are white, while $b_1$ and $b_2$ are black, {\em precisely one} of the following holds.
  • There is a path of white vertices connecting $w_1$ and $w_2$.
  • There is a path of black vertices connecting $b_1$ and $b_2$.
      Our main result is that a graph $G$ enjoys property $TRI^+(G)$ if, and only if, it is minimal with respect to property $MONO$. Building on this, we show that one can decide in polynomial time whether or not a given graph $G$ has property $MONO(G)$.
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