Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userIntertwining connectivities for vertex-minors and pivot-minors
77
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We show that for pairs $(Q,R)$ and $(S,T)$ of disjoint subsets of vertices of a graph $G$, if $G$ is sufficiently large, then there exists a vertex $v$ in $V(G)-(Qcup Rcup Scup T)$ such that there are two ways to reduce $G$ by a vertex-minor operation while preserving the connectivity between $Q$ and $R$ and the connectivity between $S$ and $T$. Our theorem implies an analogous theorem of Chen and Whittle (2014) for matroids restricted to binary matroids.
rate research
Read More
The cut-rank of a set $X$ in a graph $G$ is the rank of the $Xtimes (V(G)-X)$ submatrix of the adjacency matrix over the binary field. A split is a partition of the vertex set into two sets $(X,Y)$ such that the cut-rank of $X$ is less than $2$ and both $X$ and $Y$ have at least two vertices. A graph is prime (with respect to the split decomposition) if it is connected and has no splits. A graph $G$ is $k^{+ell}$-rank-connected if for every set $X$ of vertices with the cut-rank less than $k$, $lvert Xrvert$ or $lvert V(G)-Xrvert $ is less than $k+ell$. We prove that every prime $3^{+2}$-rank-connected graph $G$ with at least $10$ vertices has a prime $3^{+3}$-rank-connected pivot-minor $H$ such that $lvert V(H)rvert =lvert V(G)rvert -1$. As a corollary, we show that every excluded pivot-minor for the class of graphs of rank-width at most $k$ has at most $(3.5 cdot 6^{k}-1)/5$ vertices for $kge 2$. We also show that the excluded pivot-minors for the class of graphs of rank-width at most $2$ have at most $16$ vertices.
Tree-width and its linear variant path-width play a central role for the graph minor relation. In particular, Robertson and Seymour (1983) proved that for every tree~$T$, the class of graphs that do not contain $T$ as a minor has bounded path-width. For the pivot-minor relation, rank-width and linear rank-width take over the role from tree-width and path-width. As such, it is natural to examine if for every tree~$T$, the class of graphs that do not contain $T$ as a pivot-minor has bounded linear rank-width. We first prove that this statement is false whenever $T$ is a tree that is not a caterpillar. We conjecture that the statement is true if $T$ is a caterpillar. We are also able to give partial confirmation of this conjecture by proving: (1) for every tree $T$, the class of $T$-pivot-minor-free distance-hereditary graphs has bounded linear rank-width if and only if $T$ is a caterpillar; (2) for every caterpillar $T$ on at most four vertices, the class of $T$-pivot-minor-free graphs has bounded linear rank-width. To prove our second result, we only need to consider $T=P_4$ and $T=K_{1,3}$, but we follow a general strategy: first we show that the class of $T$-pivot-minor-free graphs is contained in some class of $(H_1,H_2)$-free graphs, which we then show to have bounded linear rank-width. In particular, we prove that the class of $(K_3,S_{1,2,2})$-free graphs has bounded linear rank-width, which strengthens a known result that this graph class has bounded rank-width.
A class of graphs is $chi$-bounded if there exists a function $f:mathbb Nrightarrow mathbb N$ such that for every graph $G$ in the class and an induced subgraph $H$ of $G$, if $H$ has no clique of size $q+1$, then the chromatic number of $H$ is less than or equal to $f(q)$. We denote by $W_n$ the wheel graph on $n+1$ vertices. We show that the class of graphs having no vertex-minor isomorphic to $W_n$ is $chi$-bounded. This generalizes several previous results; $chi$-boundedness for circle graphs, for graphs having no $W_5$ vertex-minors, and for graphs having no fan vertex-minors.
The extremal function $c(H)$ of a graph $H$ is the supremum of densities of graphs not containing $H$ as a minor, where the density of a graph $G$ is the ratio of the number of edges to the number of vertices. Myers and Thomason (2005), Norin, Reed, Thomason and Wood (2020), and Thomason and Wales (2019) determined the asymptotic behaviour of $c(H)$ for all polynomially dense graphs $H$, as well as almost all graphs $H$ of constant density. We explore the asymptotic behavior of the extremal function in the regime not covered by the above results, where in addition to having constant density the graph $H$ is in a graph class admitting strongly sublinear separators. We establish asymptotically tight bounds in many cases. For example, we prove that for every planar graph $H$, $$c(H) = (1+o(1))cdotmaxleft{frac{|V(H)|}{2},|V(H)| - alpha (H)right},$$ extending recent results of Haslegrave, Kim and Liu (2020). We also show that an asymptotically tight bound on the extremal function of graphs in minor-closed families proposed by Haslegrave, Kim and Liu (2020) is equivalent to a well studied open weakening of Hadwigers conjecture.
We make the first step towards a nerve theorem for graphs. Let $G$ be a simple graph and let $mathcal{F}$ be a family of induced subgraphs of $G$ such that the intersection of any members of $mathcal{F}$ is either empty or connected. We show that if the nerve complex of $mathcal{F}$ has non-vanishing homology in dimension three, then $G$ contains the complete graph on five vertices as a minor. As a consequence we confirm a conjecture of Goaoc concerning an extension of the planar $(p,q)$ theorem due to Alon and Kleitman: Let $mathcal{F}$ be a finite family of open connected sets in the plane such that the intersection of any members of $mathcal{F}$ is either empty or connected. If among any $p geq 3$ members of $mathcal{F}$ there are some three that intersect, then there is a set of $C$ points which intersects every member of $mathcal{F}$, where $C$ is a constant depending only on $p$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
