Obstruction characterization of co-TT graphs

TL;DR

This paper characterize signed-interval bigraphs and signed-interval graphs in terms of their biadjacency matrices and adjacency matrices, respectively, and concludes that this actually solve the representation characterization problem of co-TT graphs posed by Monma, Reed and Trotter.

Abstract

Threshold tolerance graphs and their complement graphs, known as co-TT graphs, were introduced by Monma, Reed, and Trotter[24]. Building on this, Hell et al.[19] introduced the concept of negative interval. Then they proceeded to define signedinterval digraphs/ bigraphs, demonstrating their equivalence to several seemingly distinct classes of digraphs/ bigraphs. They also showed that co-TT graphs are equivalent to symmetric signed-interval digraphs, where some vertices of the digraphs have loops and others do not. We have showed that this actually solve the representation characterization problem of co-TT graphs posed by Monma, Reed and Trotter [24]. In this paper, we characterize signed-interval bigraphs and signed-interval graphs in terms of their biadjacency matrices and adjacency matrices, respectively. Moreover we emphasize on the geometric representation of signed-interval graphs, i.e. co-TT graphs. Finally, by utilizing the geometric representation of signed-interval graphs, we resolve the open problem of characterizing co-TT graphs in terms of minimal forbidden induced subgraphs, a problem initially posed by Monma, Reed, and Trotter in the same paper.

Figures (6)

• Figure : Fig. 1. A co-TT graph and its co-TT representation; observe that $yv$ is an edge as $3\leq 3$ and $5\leq 6$ but $xv$ is not an edge as $5\nleq 4$.
• Figure : Fig. 2. A strongly chordal graph and $v_1$ is a simple vertex, since $N[v_1]=\{ v_1, v_2, v_4, v_5\}$ and $N[v_1]\subseteq N[v_2]\subseteq N[v_4]= N[v_5]$.
• Figure : Fig. 3. The Sun graph $S_4$
• Figure : Fig. 4. These chordal graphs are not interval graphs because each of them contains an asteroidal triple (AT) of vertices. Additionally, each of the two lower graphs contains at least six vertices.
• Figure : Fig. 5. $(i)$ A stair partition $(L,U)$, $(ii)$ A Ferrers matrix $F$, $(iii)$ The complement of the Ferrers matrix $F$.

Theorems & Definitions (26)

• Theorem 1
• Theorem 2: 8
• Theorem 3: 22
• Theorem 4: 22
• Theorem 5: 27
• Theorem 6: 21
• Theorem 7
• Theorem 8
• Theorem 9: 9
• Theorem 10: 12