On strict Semibound Graphs of Posets

Abstract

For a poset $P=(V(P),{\le }_P)$, the strict semibound graph of $P$ is the graph $ssb(P)$ on $V(ssb(P))=V(P)$ for which vertices $u$ and $v$ of $ssb(P)$ are adjacent if and only if $u\ne v$ and there exists an element $x\in V(P)$ distinct from $u$ and $v$ such that $x{\le }_{P}u,v$ or $u,v{\le }_{P}x$. We prove that a poset $P$ is connected if and only if the induced subgraph $⟨max(P){⟩}_{ssb(P)}$ is connected. We also characterize posets whose strict semibound graphs are triangle-free.