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A two-block pure blockade can realize equality in the additive kappa theorem

Example

Equality can occur in the additive $κ$ theorem for a two-block pure blockade.

Facts & Assumptions

Given: The complete graph $K_{5}$ on vertices $a_{1}, a_{2}, b_{1}, b_{2}, b_{3}$ , with blocks $A : = {a_{1}, a_{2}}$ and $B : = {b_{1}, b_{2}, b_{3}}$ .

[L1]

The pattern graph of a pure blockade records an edge exactly when the two corresponding blocks are complete (The pattern graph of a pure blockade).

[L2]

A clique on $m$ vertices has $ω = m$ , $α = 1$ , and therefore $κ = m$ (Cliques, stable sets, the clique number $ω (G)$ and stability number $α (G)$ , The parameter kappa(G)=alpha(G)omega(G)).

[L3]

The additive $κ$ theorem states that a pure blockade with cograph pattern satisfies $κ (G [V (B)]) \geq \sum_{i} κ (G [B_{i}])$ (A pure blockade with a cograph pattern has additive kappa).

Verification

technique · direct computation

1.1

The blockade $(A, B)$ is pure, and the two blocks are complete to each other because the ambient graph is $K_{5}$ . Hence its pattern graph is $K_{2}$ , which is a cograph.

L1given

1.2

The induced subgraphs on $A$ , on $B$ , and on $A \cup B$ are respectively $K_{2}$ , $K_{3}$ , and $K_{5}$ . By [L2], $κ (G [A]) = 2, κ (G [B]) = 3, κ (G [A \cup B]) = 5.$ So $κ (G [A \cup B]) = κ (G [A]) + κ (G [B]) .$

2.1

Thus this two-block pure blockade attains equality in the inequality from [L3].

step 1.1step 1.2L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Maria Chudnovsky, Alex Scott, Paul Seymour, and Sophie Spirkl, Erdos-Hajnal for graphs with no 5-hole, Theorem 5.1 (standard reference, not scraped)