Bijective Homomorphism Research Articles

Sofic profiles of $$S(\omega )$$ and computability

We show that for every sofic chunk $E$ there is a bijective homomorphism $f: E_c \rightarrow E$, where $E_c$ is a chunk of the group of computable permutations of $\mathbb{N}$ so that the approximating morphisms of $E$ can be viewed as restrictions of permutations of $E_c$ to finite subsets of $\mathbb{N}$. Using this we study some relevant effectivity conditions associated with sofic chunks and their profiles.

Subexponential algorithms for variants of the homomorphism problem in string graphs

We consider subexponential algorithms finding weighted homomorphisms from intersection graphs of curves (string graphs) with n vertices to a fixed graph H. We provide a complete dichotomy: if H has no two vertices sharing two common neighbors, then the problem can be solved in time 2O(n2/3log⁡n), otherwise there is no subexponential algorithm, assuming the ETH. Then we consider locally constrained homomorphisms. We show that for each target graph H, the locally injective and locally bijective homomorphism problems can be solved in time 2O(nlog⁡n) in string graphs. For locally surjective homomorphisms we show a dichotomy for H being a path or a cycle. If H is P3 or C4, then the problem can be solved in time 2O(n2/3log3/2⁡n) in string graphs; otherwise, assuming the ETH, there is no subexponential algorithm. As corollaries, we obtain new results concerning the complexity of homomorphism problems in Pt-free graphs.

Reversibility of extreme relational structures

A relational structure $${{\mathbb {X}}}$$ is called reversible iff each bijective homomorphism from $${{\mathbb {X}}}$$ onto $${{\mathbb {X}}}$$ is an isomorphism, and linear orders are prototypical examples of such structures. One way to detect new reversible structures of a given relational language L is to notice that the maximal or minimal elements of isomorphism-invariant sets of interpretations of the language L on a fixed domain X determine reversible structures. We isolate certain syntactical conditions providing that a satisfiable $$L_{\infty \omega }$$ -theory defines a class of interpretations having extreme elements on a fixed domain and detect several classes of reversible structures. For some of these classes, we characterize the corresponding reversible extreme interpretations. In particular, we characterize the reversible countable ultrahomogeneous graphs.

Reversible Disjoint Unions of Well Orders and Their Inverses

A poset $\mathbb {P}$ is called reversible iff every bijective homomorphism $f:\mathbb {P} \rightarrow \mathbb {P}$ is an automorphism. Let $\mathcal {W}$ and $\mathcal {W}^{*}$ denote the classes of well orders and their inverses respectively. We characterize reversibility in the class of posets of the form $\mathbb {P} =\bigcup _{i\in I}\mathbb {L}_{i}$, where $\mathbb {L}_{i}, i\in I$, are pairwise disjoint linear orders from $\mathcal {W} \cup \mathcal {W}^{*}$. First, if $\mathbb {L}_{i} \in \mathcal {W}$, for all i ∈ I, and $\mathbb {L}_{i} \cong \alpha _{i} =\gamma _{i}+n_{i}\in \text {Ord}$, where γi ∈Lim ∪{0} and ni ∈ ω, defining Iα := {i ∈ I : αi = α}, for α ∈Ord, and Jγ := {j ∈ I : γj = γ}, for γ ∈Lim ∪{0}, we prove that $\bigcup _{i\in I} \mathbb {L}_{i}$ is a reversible poset iff 〈αi : i ∈ I〉 is a finite-to-one sequence, that is, |Iα| < ω, for all α ∈Ord, or there exists γ = max{γi : i ∈ I}, for α ≤ γ we have |Iα| < ω, and 〈ni : i ∈ Jγ ∖ Iγ〉 is a reversible sequence of natural numbers. The same holds when $\mathbb {L}_{i} \in \mathcal {W}^{*}$, for all i ∈ I. In the general case, the reversibility of the whole union is equivalent to the reversibility of the union of components from $\mathcal {W}$ and the union of components from $\mathcal {W}^{*}$.

3-connected reduction for regular graph covers

A graph Gcovers a graph H if there exists a locally bijective homomorphism from G to H. We deal with regular coverings in which this homomorphism is prescribed by an action of a semiregular subgroup Γ of Aut(G); so H≅G∕Γ. In this paper, we study the behavior of regular graph covering with respect to 1-cuts and2-cuts in G.We describe reductions which produce a series of graphs G=G0,…,Gr such that Gi+1 is created from Gi by replacing certain inclusion minimal subgraphs with colored edges. The process ends with a primitive graphGr which is either 3-connected, or a cycle, or K2. This reduction can be viewed as a non-trivial modification of reductions of Mac Lane (1937), Trachtenbrot (1958), Tutte (1966), Hopcroft and Tarjan (1973), Cuningham and Edmonds (1980), Walsh (1982), and others. A novel feature of our approach is that in each step all essential information about symmetries of G are preserved.A regular covering projection G0→H0 induces regular covering projections Gi→Hi where Hi is the ith quotient reduction of H0. This property allows to construct all possible quotients H0 of G0 from the possible quotients Hr of Gr. By applying this method to planar graphs, we give a proof of Negami’s Theorem (1988). Our structural results are also used in subsequent papers for regular covering testing when G is a planar graph and for an inductive characterization of the automorphism groups of planar graphs (see Babai (1973) as well).

Packing bipartite graphs with covers of complete bipartite graphs

For a set S of graphs, a perfect S-packing (S-factor) of a graph G is a set of mutually vertex-disjoint subgraphs of G that each are isomorphic to a member of S and that together contain all vertices of G. If G allows a covering (locally bijective homomorphism) to a graph H, i.e., a vertex mapping f:VG→VH satisfying the property that f(u)f(v) belongs to EH whenever the edge uv belongs to EG such that for every u∈VG the restriction of f to the neighborhood of u is bijective, then G is an H-cover. For some fixed H let S(H) consist of all connected H-covers. Let Kk,ℓ be the complete bipartite graph with partition classes of size k and ℓ, respectively. For all fixed k,ℓ≥1, we determine the computational complexity of the problem that tests whether a given bipartite graph has a perfect S(Kk,ℓ)-packing. Our technique is partially based on exploring a close relationship to pseudo-coverings. A pseudo-covering from a graph G to a graph H is a homomorphism from G to H that becomes a covering to H when restricted to a spanning subgraph of G. We settle the computational complexity of the problem that asks whether a graph allows a pseudo-covering to Kk,ℓ for all fixed k,ℓ≥1.

Comparing Universal Covers in Polynomial Time

The universal cover T G of a connected graph G is the unique (possibly infinite) tree covering G, i.e., that allows a locally bijective homomorphism from T G to G. It is well-known that if a graph G covers a graph H, then their universal covers are isomorphic, and that the latter can be tested in polynomial time by checking if G and H share the same degree refinement matrix. We extend this result to locally injective and locally surjective homomorphisms by following a very different approach. Using linear programming techniques we design two polynomial time algorithms that check if there exists a locally injective or a locally surjective homomorphism, respectively, from a universal cover T G to a universal cover T H (both given by their degree matrices). This way we obtain two heuristics for testing the corresponding locally constrained graph homomorphisms. Our algorithm can also be used for testing (subgraph) isomorphism between universal covers, and for checking if there exists a locally injective or locally surjective homomorphism (role assignment) from a given tree to an arbitrary graph H.