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The concept of linear tangle was introduced as an obstruction to mixed searching number. The concept of (maximal) single ideal has been introduced as an obstruction to linear-width. Moreover, it was already known that mixed search number is equivalent to linear-width. Hence, by combining those result s , we obtain a proof of the equivalence between linear tangle and maximal single ideal. This short report gives an alternative proof of the equivalence.

A graph searching game is a game where searchers (or cops) want to capture a fugitive (or robber) and the fugitive want to escape from the searchers, and they move through a graph for their purpose. Graph searching games have been well-studied [

There are several variants of graph searching games such as edge search, node search, and mixed search (see e.g. [

The concept of linear tangle was introduced in [

The concept of (maximal) single ideal has been introduced in [

In this paper, we consider a pair ( E , f ) rather than graphs, where E is an underlying set and f is a symmetric submodular function on E, and such a pair is called connectivity system (see cf. [

A function f : 2 E → Z is symmetric submodular if f satisfies the following:

1) f ( X ) = f ( X ¯ ) for any X ⊆ E ,

2) f ( X ) + f ( Y ) ≥ f ( X ∪ Y ) + f ( Y ∩ X ) for any X , Y ⊆ E .

It is known that a symmetric submodular function f satisfies the following properties [

1) f ( X ) ≥ f ( ∅ ) and

2) f ( X ) + f ( Y ) ≥ f ( X \ Y ) + f ( Y \ X ) .

A set X is k-efficient if f ( X ) ≤ k . Throughout the paper, f means a symmetric submodular function, k a fixed positive integer, and we assume that f ( { e } ) ≤ k for every e ∈ E , hence we have f ( ∅ ) ≤ k .

Definition 1 ( [

(L1) ∅ ∈ L ,

(L2) For each k-efficient subset X of E, exactly one of X or X ¯ in L ,

(L3) If X , Y ∈ L , e ∈ E , and f ( { e } ) ≤ k , then X ∪ Y ∪ { e } ≠ E holds.

Definition 2 ( [

(S1) E ∉ M ,

(S2) For each k-efficient subset A of E, exactly one of A or A ¯ in M ,

(S3) If A , B ⊆ E , A ⊂ B , B ∈ M , and f ( A ) ≤ k , then A ∈ M holds,

(S4) If A ∈ M , e ∈ E , f ( { e } ) ≤ k , and f ( A ∪ { e } ) ≤ k , then A ∪ { e } ∈ M holds.

Since we assume that f ( { e } ) ≤ k for every e ∈ E , the axioms (L3) and (S4) can be restated, respectively, as follows:

(L3) If X , Y ∈ L and e ∈ E , then X ∪ Y ∪ { e } ≠ E holds.

(S4) If A ∈ M , e ∈ E , and f ( A ∪ { e } ) ≤ k , then A ∪ { e } ∈ M holds.

Lemma 1. A linear tangle L of order k + 1 is a maximal single ideal of order k + 1 .

Proof. From the axioms (L1) and (L2), it is obvious that L satisfies the axioms (S1) and (S2).

We claim that L satisfies the axioms (S3). Suppose to, the contrary, that there exist k-efficient subsets A and B such that A ⊆ B , B ∈ L , and A ∉ L . Then, we have A ¯ ∈ L by (L2), and for any e ∈ E , A ¯ ∪ { e } ∪ B = E holds, but this contradicts the axiom (L3).

Finally, we show that L satisfies the axioms (S4). Suppose to, the contrary, that there exists k-efficient subset A ∈ L and an element e ∈ E such that f ( A ∪ { e } ) ≤ k and A ∪ { e } ∉ L hold. Then, we have A ∪ { e } ¯ ∈ L , hence A , A ∪ { e } ¯ ∈ L and A ∪ A ∪ { e } ¯ ∪ { e } = E hold, however, this contradicts the axiom (L3).

Lemma 2. A maximal single ideal M of order k + 1 is a linear tangle of order k + 1 .

Proof. From the axioms (S1) and (S2), it is obvious that M satisfies the axioms (L1) and (L2).

We show that M satisfies the axiom (L3). Suppose to, the contrary, that there exists a triple ( X , Y , { e } ) such that X , Y ∈ M , e ∈ E , and X ∪ Y ∪ { e } = E . We choose a triple minimizing | X ∩ Y | in such triples ( X , Y , { e } ) . First, we claim that X ∩ Y = ∅ holds. From the axiom (S3) and the submodularity, 2 k ≥ f ( X ) + f ( Y ) ≥ f ( X \ Y ) + f ( Y \ X ) holds. Thus, at least one of f ( X \ Y ) or f ( Y \ X ) is at most k. Without loss of generality, we may assume that f ( X \ Y ) is at most k. Hence, by (S3) from X \ Y ⊆ X , we have X \ Y ∈ M . If X ∩ Y ≠ ∅ , then we have | X ∩ Y | > | ( X \ Y ) ∩ Y | , however, this contradicts the choice of the triple. Thus, we have shown that X ∩ Y = ∅ .

Next, we claim that e ∉ X holds. Suppose if not, that is, if e ∈ X , then we have X ∪ Y = E , which implies that X ¯ = Y ∈ M holds, however, this contradicts the axiom (S2). Similarly, we know that e ∉ Y holds.

Now, we know that the triple ( X , Y , { e } ) consists of a partition of E. Hence, we have f ( X ∪ { e } ) = f ( Y ¯ ) = f ( Y ) ≤ k , and from this, it follows that X ∪ { e } = Y ¯ ∈ M holds by the axiom (S4). However, this contradicts the axiom (S2).

From lemmas 1 and 2, we have the following theorem.

Theorem 1. Under the assumption that f ( { e } ) ≤ k for every e ∈ E , F is a linear tangle of order k + 1 iff F is a maximal single ideal of order k + 1 .

This work was supported by JSPS KAKENHI Grant Number 15K00007.

The authors declare no conflicts of interest regarding the publication of this paper.

Fujita, T. and Yamazaki, K. (2019) Equivalence between Linear Tangle and Maximal Single Ideal. Open Journal of Discrete Mathematics, 9, 7-10. https://doi.org/10.4236/ojdm.2019.91002