The path-plus-isolate co-Bird obstruction

The Bird graph and co-Bird supplies the following definition: The Bird graph is the graph on vertices $\{x_1,x_2,x_3,y,z,w\}$ with edge set $\{x_1{x}_2,x_2{x}_3,x_1{x}_3,x_{1}y,x_{2}z,yw\}\mathrm{.}$ So $\{x_1,x_2,x_3,y,z\}$ spans the bull, and $w$ is a new leaf attached to the horn vertex $y$. The co-Bird graph is the complement of the Bird graph.

Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs supplies the following definition: Let $G$ be a finite simple graph and let $A,B\subseteq V(G)$ be disjoint. An edge between $A$ and $B$ is an edge $ab$ with $a\in A$ and $b\in B$. The pair $(A,B)$ is: - complete when every $a\in A$ is adjacent to every $b\in B$; - anticomplete when no $a\in A$ is adjacent to any $b\in B$; - pure when it is complete or anticomplete; and - mixed when it is neither complete nor anticomplete. Adjacency is the symmetric edge relation of $G$ (def-finite-simple-graph, def-graph-adjacency-incidence-neighbourhood-and-degree). If $A=\mathrm{\varnothing }$ or $B=\mathrm{\varnothing }$, the pair is both complete and anticomplete, hence pure and not mixed.

$H$-free and $\mathcal{F}$-free graphs under the induced-subgraph convention supplies the following definition: For finite graphs $H$ and $G$, the graph $G$ is $H$-free when $G$ has no induced copy of $H$ (def-induced-embedding-and-induced-copy). Equivalently, ${\mathrm{ind}}_H(G)=0$ (def-induced-copy-number). For a family $\mathcal{F}$ of finite graphs, a finite graph $G$ is $\mathcal{F}$-free when it is $H$-free for every $H\in \mathcal{F}$. Throughout this page, “free” always refers to induced subgraphs. It does not merely prohibit ordinary subgraph copies.