Defining Recursive Predicates in Graph Orders

09/10/2017
by   Ramanathan S. Thinniyam, et al.
0

We study the first order theory of structures over graphs i.e. structures of the form (G,τ) where G is the set of all (isomorphism types of) finite undirected graphs and τ some vocabulary. We define the notion of a recursive predicate over graphs using Turing Machine recognizable string encodings of graphs. We introduce the notion of a capable structure over graphs, which is one satisfying the conditions : (1) definability of arithmetic, (2) definability of cardinality of a graph, and (3) definability of two particular graph predicates related to vertex labellings of graphs. We then show any capable structure can define every recursive predicate over graphs. We identify capable structures which are expansions of graph orders, which are structures of the form (G,≤) where ≤ is a partial order. We show that the subgraph order i.e. (G,≤_s), induced subgraph order with one constant P_3 i.e. (G,≤_i,P_3) and an expansion of the minor order for counting edges i.e. (G,≤_m,sameSize(x,y)) are capable structures. In the course of the proof, we show the definability of several natural graph theoretic predicates in the subgraph order which may be of independent interest. We discuss the implications of our results and connections to Descriptive Complexity.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
05/01/2023

Embeddability of graphs and Weihrauch degrees

We study the complexity of the following related computational tasks con...
research
10/01/2022

Conditions for minimally tough graphs

Katona, Soltész, and Varga showed that no induced subgraph can be exclud...
research
12/14/2017

Structural and computational results on platypus graphs

A platypus graph is a non-hamiltonian graph for which every vertex-delet...
research
08/02/2022

Extremal numbers of disjoint triangles in r-partite graphs

For two graphs G and F, the extremal number of F in G, denoted by ex(G,F...
research
11/30/2021

Induced betweenness in order-theoretic trees

The ternary relation B(x,y,z) of betweenness states that an element y is...
research
05/08/2023

Isomorphisms between dense random graphs

We consider two variants of the induced subgraph isomorphism problem for...
research
06/28/2018

Graphs without 2-community structures

In the context of community structure detection, we study the existence ...

Please sign up or login with your details

Forgot password? Click here to reset