50 research outputs found
Cores of Countably Categorical Structures
A relational structure is a core, if all its endomorphisms are embeddings.
This notion is important for computational complexity classification of
constraint satisfaction problems. It is a fundamental fact that every finite
structure has a core, i.e., has an endomorphism such that the structure induced
by its image is a core; moreover, the core is unique up to isomorphism. Weprove
that every \omega -categorical structure has a core. Moreover, every
\omega-categorical structure is homomorphically equivalent to a model-complete
core, which is unique up to isomorphism, and which is finite or \omega
-categorical. We discuss consequences for constraint satisfaction with \omega
-categorical templates
List covering of regular multigraphs
A graph covering projection, also known as a locally bijective homomorphism,
is a mapping between vertices and edges of two graphs which preserves
incidencies and is a local bijection. This notion stems from topological graph
theory, but has also found applications in combinatorics and theoretical
computer science.
It has been known that for every fixed simple regular graph of valency
greater than 2, deciding if an input graph covers is NP-complete. In recent
years, topological graph theory has developed into heavily relying on multiple
edges, loops, and semi-edges, but only partial results on the complexity of
covering multigraphs with semi-edges are known so far. In this paper we
consider the list version of the problem, called \textsc{List--Cover}, where
the vertices and edges of the input graph come with lists of admissible
targets. Our main result reads that the \textsc{List--Cover} problem is
NP-complete for every regular multigraph of valency greater than 2 which
contains at least one semi-simple vertex (i.e., a vertex which is incident with
no loops, with no multiple edges and with at most one semi-edge). Using this
result we almost show the NP-co/polytime dichotomy for the computational
complexity of \textsc{ List--Cover} of cubic multigraphs, leaving just five
open cases.Comment: Accepted to IWOCA 202
On a stronger reconstruction notion for monoids and clones
Motivated by reconstruction results by Rubin, we introduce a new
reconstruction notion for permutation groups, transformation monoids and
clones, called automatic action compatibility, which entails automatic
homeomorphicity. We further give a characterization of automatic
homeomorphicity for transformation monoids on arbitrary carriers with a dense
group of invertibles having automatic homeomorphicity. We then show how to lift
automatic action compatibility from groups to monoids and from monoids to
clones under fairly weak assumptions. We finally employ these theorems to get
automatic action compatibility results for monoids and clones over several
well-known countable structures, including the strictly ordered rationals, the
directed and undirected version of the random graph, the random tournament and
bipartite graph, the generic strictly ordered set, and the directed and
undirected versions of the universal homogeneous Henson graphs.Comment: 29 pp; Changes v1-->v2::typos corr.|L3.5+pf extended|Rem3.7 added|C.
Pech found out that arg of L5.3-v1 solved Probl2-v1|L5.3, C5.4, Probl2 of v1
removed|C5.2, R5.4 new, contain parts of pf of L5.3-v1|L5.2-v1 is now
L5.3,merged with concl of C5.4-v1,L5.3-v2 extends C5.4-v1|abstract, intro
updated|ref[24] added|part of L5.3-v1 is L2.1(e)-v2, another part merged with
pf of L5.2-v1 => L5.3-v
First-order limits, an analytical perspective
In this paper we present a novel approach to graph (and structural) limits
based on model theory and analysis. The role of Stone and Gelfand dualities is
displayed prominently and leads to a general theory, which we believe is
naturally emerging. This approach covers all the particular examples of
structural convergence and it put the whole in new context. As an application,
it leads to new intermediate examples of structural convergence and to a "grand
conjecture" dealing with sparse graphs. We survey the recent developments
Permutation monoids and MB-homogeneity for graphs and relational structures
In this paper we investigate the connection between infinite permutation monoids and bimorphism monoids of first-order structures. Taking our lead from the study of automorphism groups of structures as infinite permutation groups and the more recent developments in the field of homomorphism-homogeneous structures, we establish a series of results that underline this connection. Of particular interest is the idea of MB-homogeneity; a relational structure M is MB-homogeneous if every monomorphism between finite substructures of M extends to a bimorphism of M. The results in question include a characterisation of closed permutation monoids, a Fraisse-like theorem for MB-homogeneous structures, and the construction of 2â”0 pairwise non-isomorphic countable MB-homogeneous graphs. We prove that any finite group arises as the automorphism group of some MB-homogeneous graph and use this to construct oligomorphic permutation monoids with any given finite group of units. We also consider MB-homogeneity for various well-known examples of homogeneous structures and in particular give a complete classification of countable homogeneous undirected graphs that are also MB-homogeneous
Homomorphisms of (j,k)-mixed graphs
A mixed graph is a simple graph in which a subset of the edges have been assigned directions to form arcs. For non-negative integers j and k, a (j,k)âmixed graph is a mixed graph with j types of arcs and k types of edges. The collection of (j,k)âmixed graphs contains simple graphs ((0,1)âmixed graphs), oriented graphs ((1,0)âmixed graphs) and kâedge- coloured graphs ((0,k)âmixed graphs).A homomorphism is a vertex mapping from one (j,k)âmixed graph to another in which edge type is preserved, and arc type and direction are preserved. The (j,k)âchromatic number of a (j,k)âmixed graph is the least m such that an mâcolouring exists. When (j,k)=(0,1), we see that these definitions are consistent with the usual definitions of graph homomorphism and graph colouring.In this thesis we study the (j,k)âchromatic number and related parameters for different families of graphs, focussing particularly on the (1,0)âchromatic number, more commonly called the oriented chromatic number, and the (0,k)âchromatic number.In addition to considering vertex colourings, we also consider incidence colourings of both graphs and digraphs. Using systems of distinct representatives, we provide a new characterisation of the incidence chromatic number. We define the oriented incidence chromatic number and find, by way of digraph homomorphism, a connection between the oriented incidence chromatic number and the chromatic number of the underlying graph. This connection motivates our study of the oriented incidence chromatic number of symmetric complete digraphs.Un graphe mixte est un graphe simple tel que un sous-ensemble des arĂȘtes a une orientation. Pour entiers non nĂ©gatifs j et k, un graphe mixte-(j,k) est un graphe mixte avec j types des arcs and k types des arĂȘtes. La famille de graphes mixte-(j,k) contient graphes simple, (graphes mixteâ(0,1)), graphes orientĂ© (graphes mixteâ(1,0)) and graphe colorĂ© arĂȘte âk (graphes mixteâ(0,k)).Un homomorphisme est un application sommet entre graphes mixteâ(j,k) que tel les types des arĂȘtes sont conservĂ©s et les types des arcs et leurs directions sont conservĂ©s. Le nombre chromatiqueâ(j,k) dâun graphe mixteâ(j,k) est le moins entier m tel quâil existe un homomorphisme Ă une cible avec m sommets. Quand on observe le cas de (j,k) = (0,1), on peut dĂ©terminer ces dĂ©finitions correspondent Ă les dĂ©finitions usuel pour les graphes.Dans ce mĂ©moire on etude le nombre chromatiqueâ(j,k) et des paramĂštres similaires pour diverses familles des graphes. Aussi on etude les coloration incidence pour graphes and digraphs. On utilise systĂšmes de reprĂ©sentants distincts et donne une nouvelle caractĂ©risation du nombre chromatique incidence. On define le nombre chromatique incidence orientĂ© et trouves un connexion entre le nombre chromatique incidence orientĂ© et le nombre chromatic du graphe sous-jacent