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I am working on giving an MSOL formula for an NP-hard problem; this proves that the problem is linear-time solvable on bounded treewidth graphs. Given a graph $G = (V, E)$, the problem would be to find a subset $S \subseteq V$ such that each vertex of degree at least three has at least two neighbors and at most three neighbours in $S$. The MSOL formula for the problem is:
$\exists S \subseteq V:\forall v \in V\setminus d(v) \leq 2: ((\exists {u_1, u_2} \in S) \land (u_1 \in N(v)) \land (u_2 \in N(v)))$ $\land ((\exists u_3 \in S) \land ((u_3 \in N(v) \lor (u_3 \notin N(v))))$ $\land ((\forall u \in S \setminus \{u_1,u_2, u_3\}) \land (u \notin N(v)))$ where $d(v)$ indicates the degree of $v$.

  1. Is this a valid MSOL formula? can i make use of $d(v)$ in the formula?
  2. If not, how can i modify it in such a way that the formula solves the problem?
Balchandar Reddy
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1 Answers1

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main concern is whether or not $d(v)$ can be used in the formula

The answer to this is no.

However, if you want to ask whether the degree is (equal to, at least, at most) some fixed constant $c$, then you can make a formula like this:

$$d(v) \geq c \; \triangleq \; \exists u_1 \exists u_2 \cdots \exists u_c \; vu_1 \in E \land vu_2 \in E \land \cdots \land vu_c \in E \land u_1 \neq u_2 \land \cdots \land u_{c-1} \neq u_c$$

Ainsley H.
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