Connectedness of independence attractors of graphs with independence number three

An independent set in a simple graph $G$ is a set of pairwise non-adjacent vertices in $G$. The independence polynomial of $G$, denoted by $I_G$ is defined as $1 + a_1 z + a_2 z^2+\cdots+a_d z^{d}$, where $a_i$ denotes the number of independent sets with cardinality $i$ and $d$ is the cardinality of a largest independent set in $G$. This $d$ is known as the independence number of $G$. Let $G^m$ denote the $m$-times lexicographic product of $G$ with itself. The independence attractor of $G$, denoted by $\mathcal{A}(G)$ is defined as $\mathcal{A}(G) = \lim\limits_{m\rightarrow \infty} \{z: I_{G^m}(z)=0\}$, where the limit is taken with respect to the Hausdorff metric defined on the space of all compact subsets of the plane. This paper investigates the connectedness of the independence attractors of all graphs with independence number three. Let the independence polynomial of $G$ be $1+a_1 z +a_2 z^2 +a_3 z^3$. For $a_1 =3$, $\mathcal{A}(G)$ turns out to be $\{-1\} \cup \{z: |z+1|=1\}$. For $a_1 >3$, we prove the following. If $a_2 ^2 \leq 3 a_1 a_3$, or $3 a_1 a_3 < a_2 ^2 < 4a_3 (a_1 -1)$ then $\mathcal{A}(G)$ is totally disconnected. For $a_2 ^2 =4a_3 (a_1 -1)$, $\mathcal{A}(G)$ is connected when $a_1 =5$ and is disconnected but not totally disconnected for all other values of $a_1$. If $a_2 ^2 > 4a_3 (a_1 -1)$ then $\mathcal{A}(G)$ can be connected, totally disconnected or disconnected but not totally disconnected depending on further conditions involving $a_1, a_2$ and $a_3$. Examples of graphs exhibiting all the possibilities are provided.