Return to Answer

Replaced earlier answer, which used an incorrect definition of interval filament graphs.

Unipolar graphs are interval filaments graphs. Brief sketch: If $G$ is unipolar there are vertex sets $A$ and $B$ s.th. $A$ induces a clique and $B$ induces a set of cliques with vertex sets $B_1, \ldots, B_k$, say.

Each filament has two endpoints on the x-axis, and we order the left endpoints to be consistent with the grouping $A, B_1, \ldots, B_k$, and then place the right endpoints in the reverse order to the right of $B_k$.

Because the right endpoints are in the reverse order of the left endpoints, we can draw all filaments simultaneously as $/\backslash$-shaped curves and so that no two of the filaments intersect. To create an intersection between an $A$-filament $a$ and a $B_i$-filament $b$, let $x$ be a coordinate inside the group $B_i$, where both $a$ and $b$ are present. We can then push $a$ up (close to $x$) until it is above all the other $B_i$-filaments at $x$, and push $a$ down, until it it below all the other $A$-filaments at $x$, and then make $a$ and $b$ intersect. Creating intersections between two filaments belonging to the same group is similar (and easier).

Replaced earlier answer, which used an incorrect definition of interval filament graphs.

Unipolar graphs are interval filaments graphs. Brief sketch: If $G$ is unipolar there are vertex sets $A$ and $B$ s.th. $A$ induces a clique and $B$ induces a set of cliques with vertex sets $B_1, \ldots, B_k$, say.

Each filament has two endpoints on the x-axis, and we order the endpoints along the $x$-axis as $A, B_1, B^R_1, B_2, B^R_2, \ldots, B_k, B^R_k, A^R$, where we think of $A$, and $B_i$ as ordered sets, and let $X^R$ denote the reverse of the ordered set $X$.

We can then draw all filaments simultaneously as $/\backslash$-shaped curves and so that no two of the filaments intersect. To create an intersection between an $A$-filament $a$ and a $B_i$-filament $b$, let $x$ be a coordinate inside the group $B_i$, where both $a$ and $b$ are present. We can then push $a$ up (close to $x$) until it is above all the other $B_i$-filaments at $x$, and push $a$ down, until it it below all the other $A$-filaments at $x$, and then make $a$ and $b$ intersect. Creating intersections between two filaments belonging to the same group is similar (and easier).

Unipolar graphs are interval filaments graphs. Brief sketch: If $G$ is unipolar there are vertex sets $A$ and $B$ s.th. $A$ induces a clique and $B$ induces a set of cliques with vertex sets $B_1, \dotsc, B_k$, say. Create a filament for each vertex of $G$ attached to the $x$-axis grouping them in order $B_1, \dotsc, B_k, A$. We can entangle the filaments in each group to create the cliques $A, B_1, \dotsc, B_k$ in $G$. Any edge between $A$ and $B_k$ can then be created by having the filaments of their endpoints intersect close to the boundary between $B_k$ and $A$. We place the intersections above all earlier intersections to avoid prohibited intersections. Once $B_k$ is done, we continue with edges between $B_{k-1}$ and $A$. And so on.

Replaced earlier answer, which used an incorrect definition of interval filament graphs.

Unipolar graphs are interval filaments graphs. Brief sketch: If $G$ is unipolar there are vertex sets $A$ and $B$ s.th. $A$ induces a clique and $B$ induces a set of cliques with vertex sets $B_1, \ldots, B_k$, say.

Each filament has two endpoints on the x-axis, and we order the left endpoints to be consistent with the grouping $A, B_1, \ldots, B_k$, and then place the right endpoints in the reverse order to the right of $B_k$.

Because the right endpoints are in the reverse order of the left endpoints, we can draw all filaments simultaneously as $/\backslash$-shaped curves and so that no two of the filaments intersect. To create an intersection between an $A$-filament $a$ and a $B_i$-filament $b$, let $x$ be a coordinate inside the group $B_i$, where both $a$ and $b$ are present. We can then push $a$ up (close to $x$) until it is above all the other $B_i$-filaments at $x$, and push $a$ down, until it it below all the other $A$-filaments at $x$, and then make $a$ and $b$ intersect. Creating intersections between two filaments belonging to the same group is similar (and easier).

Unipolar graphs are interval filaments graphs. Brief sketch: If G is unipolar there are vertex sets A and B s.th A induces a clique and B induces a set of cliques with vertex sets B1, .., Bk, say. Create a filament for each vertex of G attached to the x-axis grouping them in order B1, ..., Bk, A. We can entangle the filaments in each group to create the cliques A, B1, ..., Bk in G. Any edge between A and Bk can then be created by having the filaments of their endpoints intersect close to the boundary between Bk and A. We place the intersections above all earlier intersections to avoid prohibited intersections. Once Bk is done, we continue with edges between B_(k-1) and A. And so on.

Unipolar graphs are interval filaments graphs. Brief sketch: If $G$ is unipolar there are vertex sets $A$ and $B$ s.th. $A$ induces a clique and $B$ induces a set of cliques with vertex sets $B_1, \dotsc, B_k$, say. Create a filament for each vertex of $G$ attached to the $x$-axis grouping them in order $B_1, \dotsc, B_k, A$. We can entangle the filaments in each group to create the cliques $A, B_1, \dotsc, B_k$ in $G$. Any edge between $A$ and $B_k$ can then be created by having the filaments of their endpoints intersect close to the boundary between $B_k$ and $A$. We place the intersections above all earlier intersections to avoid prohibited intersections. Once $B_k$ is done, we continue with edges between $B_{k-1}$ and $A$. And so on.