Obstruction characterization of co-TT graphs

Threshold tolerance graphs and their complement graphs ( known as co-TT graphs) were introduced by Monma, Reed and Trotter[24]. Introducing the concept of negative interval Hell et al.[19] defined signed-interval bigraphs/digraphs and have shown that they are equivalent to several seemingly different classes of bigraphs/digraphs. They have also shown that co-TT graphs are equivalent to symmetric signed-interval digraphs. In this paper we characterize signed-interval bigraphs and signed-interval graphs respectively in terms of their biadjacency matrices and adjacency matrices. Finally, based on the geometric representation of signed-interval graphs we have setteled the open problem of forbidden induced subgraph characterization of co-TT graphs posed by Monma, Reed and Trotter in the same paper.Comment: arXiv admin note: substantial text overlap with arXiv:2206.0591

L(p,q)L(p,q)-Labeling of Graphs with Interval Representations

We provide upper bounds on the $L(p,q)$-labeling number of graphs which have interval (or circular-arc) representations via simple greedy algorithms. We prove that there exists an $L(p,q)$-labeling with span at most $\max\{2(p+q-1)\Delta-4q+2, (2p-1)\mu+(2q-1)\Delta-2q+1\}$ for interval $k$-graphs, $\max\{p,q\}\Delta$ for interval graphs, $\max\{p,q\}\Delta+p\omega$ for circular arc graphs, $2(p+q-1)\Delta-2q+1$ for permutation graphs and $(2p-1)\Delta+(2q-1)(\mu-1)$ for cointerval graphs. In particular, these improve existing bounds on $L(p,q)$-labeling of interval and circular arc graphs and $L(2,1)$-labeling of permutation graphs. Furthermore, we provide upper bounds on the coloring of the squares of aforementioned classes