We prove that for any class of finite or countable graphs closed under taking subgraphs, and such that every finite graph in the class has a number of edges linearly bounded in terms of the number of vertices, the edge set quantifications can be replaced by vertex set quantifications in Monadic secondorder formulas. This result extends to hypergraphs of bounded rank.

Let C be the claw K 1;3 and N the net, i.e. the only connected graph with degree sequence 333111. It is known Bedrossian 1991; Faudree and Gould 1997] that if X; Y is a pair of connected graphs such that, for any 2-connected graph G, G being X Y-free implies G is hamiltonian, then X is the claw C and Y belongs to a nite list of graphs, one of them being the net N. For any such pair X; Y we show...

A conjecture by D. Seese states that if a set of graphs has a decidable monadic second-order theory, then it is the image of a set of trees under a transformation of relational structures defined by monadic second-order formulas, or equivalently, has bounded clique-width. We prove that this conjecture is true if and only if it is true for the particular cases of bipartite undirected graphs, of ...

Courcelle, B., The monadic second-order logic of graphs V: on closing the gap between definability and recognizability, Theoretical Computer Science 80 (1991) 153-202. Context-free graph-grammars are considered such that, in every generated graph (3, a derivation tree of G can be constructed by means of monadic second-order formulas that specify its nodes, its labels, the successors of a node e...

Effect of CO2-related acidification on aspects of the larval development of the European lobster, Homarus gammarus (L.) K. E. Arnold, H. S. Findlay, J. I. Spicer, C. L. Daniels, and D. Boothroyd National Lobster Hatchery, South Quay, Padstow, Cornwall, PL28 8BL, UK Plymouth Marine Laboratory, Prospect Place, West Hoe, Plymouth, Devon, PL1 3DH, UK Marine Biology and Ecology Research Centre, Scho...

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We prove that we can specify by formulas of monadic second-order logic the unique planar embedding of a 3-connected planar graph. If the planar graph is not 3-connected but given with a linear order of its set of edges, we can also define a planar embedding by monadic second-order formulas. We cannot do so in general without the ordering, even for 2-connected planar graphs. The planar embedding...

let $gamma_{n,kappa}$ be the class of all graphs with $ngeq3$ vertices and $kappageq2$ vertex connectivity. denote by $upsilon_{n,beta}$ the family of all connected graphs with $ngeq4$ vertices and matching number $beta$ where $2leqbetaleqlfloorfrac{n}{2}rfloor$. in the classes of graphs $gamma_{n,kappa}$ and $upsilon_{n,beta}$, the elements having maximum augmented zagreb index are determined.

for a graph g, let d(g) be the set of all strong orientations of g. the orientationnumber of g is min {d(d) |d belongs to d(g)}, where d(d) denotes the diameterof the digraph d. in this paper, we determine the orientation number for some subgraphs of complete bipartite graphs.

This article establishes that the split decomposition of graphs introduced by Cunnigham, is definable in Monadic Second-Order Logic.This result is actually an instance of a more general result covering canonical graph decompositions like the modular decomposition and the Tutte decomposition of 2-connected graphs into 3-connected components. As an application, we prove that the set of graphs hav...