Abstract This chapter considers the partial order on graphs induced by the existence of homomorphisms. This order is rich enough to represent all countable partial orders. It discusses antichains in the homomorphism order, i.e., collections of incomparable graphs (graphs without homomorphisms between any two of them). Of particular interest are finite maximal antichains, as their structure turns out to be surprisingly revealing. Graphs only have trivial finite maximal antichains, while digraphs have many such antichains, of all possible sizes, arising from duality relationships. This chapter also contains the (probabilistic) proof of the sparse incomparability lemma, of the fact that asymptotically almost all graphs on n vertices are cores, and of the fact that the number of incomparable graphs on n vertices differs little (asymptotically) from the total number of nonisomorphic graphs on n vertices. The density of the homomorphism order is related to duality relationships, revealing an unexpected connection between these two seemingly unrelated concepts. Finally, it is argued that one can gain interesting insights into many traditional graph topics, such as, say, Hadwiger’s conjecture, when interpreting them as statements about the homomorphism order.

Given a family of graphs ℱ, we define a graph G to be fully ℱ-colourable if G admits a full homomorphism to some F in ℱ. We approach the problem of determining when a graph is fully ℱ-colourable in terms of minimal forbidden induced subgraphs. We provide general results which allow to obtain the exact families of forbidden induced subgraphs for full ℱ-colouring when ℱ is among some well-known families, such as threshold, trivially perfect, split, chordal, interval and strongly chordal graphs, as well as forests. Traditionally, these questions have been studied for a single graph H, not a family. Motivated by our results on the family of forests, we contribute to this research by focusing on the case of a single centipede.

Strongly chordal digraphs are included in the class of chordal digraphs and generalize strongly chordal graphs and chordal bipartite graphs. They are the digraphs that admit a linear ordering of its vertex set for which their adjacency matrix does not contain the Γ matrix as a submatrix. In general, it is not clear if these digraphs can be recognized in polynomial time. We focus on multipartite tournaments with possible loops. We give a polynomial-time recognition algorithm and a forbidden induced subgraph characterization of the strong chordality for each of the following cases: tournaments with possible loops, reflexive multipartite tournaments, irreflexive bipartite tournaments, irreflexive tournaments minus one arc, and balanced digraphs. In addition, we prove that in a strongly chordal digraph the minimum size of a total dominating set equals the maximum number of disjoint in-neighborhoods, and this number can be calculated in linear time given a Γ-free ordering of the input graph.

Since the CSP dichotomy conjecture has been established, a number of other dichotomy questions have attracted interest, including one for list homomorphism problems of signed graphs. Signed graphs arise naturally in many contexts, including for instance nowhere-zero flows for graphs embedded in non-orientable surfaces. The dichotomy classification is known for homomorphisms without list restrictions, so it is surprising that it is not known, or even conjectured, if lists are present since this usually makes the classifications easier to obtain. There is however a conjectured classification, due to Kim and Siggers, in the special case of “semi-balanced” signed graphs. These authors confirmed their conjecture for the class of reflexive signed graphs. As our main result we verify the conjecture for irreflexive signed graphs. For this purpose, we prove an extension result for two-directional ray graphs which is of independent interest and which leads to an analogous extension result for interval graphs. Moreover, we offer an alternative proof for the class of reflexive signed graphs, and a direct polynomial-time algorithm in the polynomial cases where the previous algorithms used algebraic methods of general CSP dichotomy theorems. For both reflexive and irreflexive cases the dichotomy classification depends on a result linking the absence of certain structures to the existence of a special ordering. The structures are used to prove the NP-completeness and the ordering is used to design polynomial algorithms.

We study the class of bi-arc digraphs, important from two seemingly unrelated perspectives. On the one hand, they are a broad generalization of interval graphs that include other popular generalizations of interval graphs, such as co-threshold tolerance graphs and adjusted interval digraphs. On the other hand, they are precisely the digraphs that admit the so-called conservative semilattice polymorphisms, also known as min orderings or X-underbar enumerations. These digraphs are generally interesting in studying graph homomorphisms and constraint satisfaction problems. Our main result is a forbidden obstruction characterization of the class of bi-arc digraphs and a polynomial-time recognition algorithm. In addition, we show that they are precisely the digraphs that admit certain other kinds of conservative polymorphisms, thereby collapsing these polymorphism types in the class of digraphs. We complement our result by providing a complete dichotomy classification of which general relational structures have polynomial or NP-complete recognition problems for the existence of conservative semilattice polymorphisms.

The weighted Szeged index is a recent extension of the well-known Szeged index. Trees are conjectured to achieve the minimum weighted Szeged index among all graphs with a given number of vertices. In this paper, we present new tools to analyze and characterize trees with minimum weighted Szeged index. We exhibit the best trees with up to 130 vertices and use this information, together with our formal results, to propose certain conjectures on the structure of such minimal trees. In particular, we prove that they have maximum degrees at most ten, and conjecture that the right bound is six. (A recent paper of Atanasov, Furtula, and Škrekovski proves a bound of 16.) We hope these conjectures and initial results will motivate further developments on this interesting topological index.

We introduce the class of strong cocomparability graphs, as the class of reflexive graphs whose adjacency matrix can be rearranged by a simultaneous row and column permutation to avoid the submatrix with rows 01, 10, which we call Slash. We provide an ordering characterization, a forbidden structure characterization, and a polynomial-time recognition algorithm, for the class. These results complete the picture in which in addition to, or instead of, the Slash matrix one forbids the Gamma matrix (which has rows 11, 10). It is well known that in these two cases one obtains the class of interval graphs, and the class of strongly chordal graphs, respectively. By complementation, we obtain the class of strong comparability graphs, whose adjacency matrix can be rearranged by a simultaneous row and column permutation to avoid the two-by-two identity submatrix. Thus our results give characterizations and algorithms for this class of irreflexive graphs as well. In other words, our results may be interpreted as solving the following problem: given a symmetric 0,1-matrix with 0-diagonal, can the rows and columns of be simultaneously permuted to avoid the two-by-two identity submatrix?

We consider homomorphisms of signed graphs from a computational perspective. In particular, we study the list homomorphism problem seeking a homomorphism of an input signed graph (G,σ), equipped with lists L(v)⊆V(H),v∈V(G), of allowed images, to a fixed target signed graph (H,π). The complexity of the similar homomorphism problem without lists (corresponding to all lists being L(v)=V(H)) has been previously classified by Brewster and Siggers, but the list version remains open and appears difficult. We illustrate this difficulty by classifying the complexity of the problem when H is a tree (with possible loops). The tools we develop will be useful for classifications of other classes of signed graphs, and in a future companion paper we will illustrate this by using them to classify the complexity for certain irreflexive signed graphs. The structure of the signed trees in the polynomial cases is interesting, suggesting that the class of general signed graphs for which the problems are polynomial may have nice structure, analogous to the so-called bi-arc graphs (which characterised the polynomial cases of list homomorphisms to unsigned graphs).

Each hereditary property can be characterized by its set of minimal obstructions; these sets are often unknown, or known but infinite. By allowing extra structure it is sometimes possible to describe such properties by a finite set of forbidden objects. This has been studied most intensely when the extra structure is a linear ordering of the vertex set. For instance, it is known that a graph G is k-colourable if and only if V(G) admits a linear ordering ≤ with no vertices v1≤⋯≤vk+1 such that vivi+1∈E(G) for every i∈{1,⋯,k}. In this paper, we study such characterizations when the extra structure is a circular ordering of the vertex set. We show that the classes that can be described by finitely many forbidden circularly ordered graphs include forests, circular-arc graphs, and graphs with circular chromatic number less than k. In fact, every description by finitely many forbidden circularly ordered graphs can be translated to a description by finitely many forbidden linearly ordered graphs. Nevertheless, our observations underscore the fact that in many cases the circular order descriptions are nicer and more natural.

The CSP dichotomy conjecture has been recently established, but a number of other dichotomy questions remain open, including the dichotomy classification of list homomorphism problems for signed graphs. Signed graphs arise naturally in many contexts, including for instance nowhere-zero flows for graphs embedded in non-orientable surfaces. For a fixed signed graph H , the list homomorphism problem asks whether an input signed graph G with lists L(v) ⊆ V(H), v ∈ V(G), admits a homomorphism f to H with all f(v) ∈ L(v), v ∈ V(G) . Usually, a dichotomy classification is easier to obtain for list homomorphisms than for homomorphisms, but in the context of signed graphs a structural classification of the complexity of list homomorphism problems has not even been conjectured, even though the classification of the complexity of homomorphism problems is known. Kim and Siggers have conjectured a structural classification in the special case of “weakly balanced" signed graphs. We confirm their conjecture for reflexive and irreflexive signed graphs; this generalizes previous results on weakly balanced signed trees, and weakly balanced separable signed graphs [1–3]. In the reflexive case, the result was first presented in [19], where the proof relies on a result in this paper. The irreflexive result is new, and its proof depends on first deriving a theorem on extensions of min orderings of (unsigned) bipartite graphs, which is interesting on its own.

The complexity of the list homomorphism problem for signed graphs appears difficult to classify. Existing results focus on special classes of signed graphs, such as trees [4] and reflexive signed graphs [25]. Irreflexive signed graphs are in a certain sense the heart of the problem, as noted by a recent paper of Kim and Siggers. We focus on a special class of irreflexive signed graphs, namely those in which the unicoloured edges form a spanning path or cycle, which we call separable signed graphs. We classify the complexity of list homomorphisms to these separable signed graphs; we believe that these signed graphs will play an important role for the general resolution of the irreflexive case. We also relate our results to a conjecture of Kim and Siggers concerning the special case of semi-balanced irreflexive signed graphs; we have proved the conjecture in another paper, and the present results add structural information to that topic.

Given a digraph G, a set X⊆ V(G) is said to be an absorbing set (resp. dominating set) if every vertex in the graph is either in X or is an in-neighbour (resp. out-neighbour) of a vertex in X. A set S⊆ V(G) is said to be an independent set if no two vertices in S are adjacent in G. A kernel (resp. solution) of G is an independent and absorbing (resp. dominating) set in G. The problem of deciding if there is a kernel (or solution) in an input digraph is known to be NP-complete. Similarly, the problems of computing a minimum cardinality dominating set or absorbing set or kernel, and the problems of computing a maximum cardinality independent set or kernel, are all known to be NP-hard for general digraphs. We explore the algorithmic complexity of these problems in the well known class of interval digraphs. A digraph G is an interval digraph if a pair of intervals (S_u,T_u) can be assigned to each vertex u of G such that (u,v)∈ E(G) if and only if S_u∩ T_v∅ . Many different subclasses of interval digraphs have been defined and studied in the literature by restricting the kinds of pairs of intervals that can be assigned to the vertices. We observe that several of these classes, like interval catch digraphs, interval nest digraphs, adjusted interval digraphs and chronological interval digraphs, are subclasses of the more general class of reflexive interval digraphs—which arise when we require that the two intervals assigned to a vertex have to intersect. We see as our main contribution the identification of the class of reflexive interval digraphs as an important class of digraphs. We show that while the problems mentioned above are NP-complete, and even hard to approximate, on interval digraphs (even on some very restricted subclasses of interval digraphs called point-point digraphs, where the two intervals assigned to each vertex are required to be degenerate), they are all efficiently solvable, in most of the cases linear-time solvable, in the class of reflexive interval digraphs. The results we obtain improve and generalize several existing algorithms and structural results for subclasses of reflexive interval digraphs. In particular, we obtain a vertex ordering characterization of reflexive interval digraphs that implies the existence of an O(n+m) time algorithm for computing a maximum cardinality independent set in a reflexive interval digraph, improving and generalizing the earlier known O(nm) time algorithm for the same problem for the interval nest digraphs. (Here m denotes the number of edges in the digraph not counting the self-loops.) We also show that reflexive interval digraphs are kernel-perfect and that a kernel in such digraphs can be computed in linear time. This generalizes and improves an earlier result that interval nest digraphs are kernel-perfect and that a kernel can be computed in such digraphs in O(nm) time. The structural characterizations that we show for point-point digraphs, apart from helping us construct the NP-completeness/APX-hardness reductions, imply that these digraphs can be recognized in linear time. We also obtain some new results for undirected graphs along the way: (a) We describe an O(n(n+m)) time algorithm for computing a minimum cardinality (undirected) independent dominating set in cocomparability graphs, which slightly improves the existing O(n^3) time algorithm for the same problem by Kratsch and Stewart; and (b) We show that the Red-Blue Dominating Set problem, which is NP-complete even for planar bipartite graphs, is linear-time solvable on interval bigraphs, which is a class of bipartite (undirected) graphs closely related to interval digraphs.

For efficient design of parallel algorithms on multiprocessor architectures with memory banks, simultaneous access to a specified subgraph of a graph data structure by multiple processors requires that the data items belonging to the subgraph reside in distinct memory banks. Such "conflict-free" access to parallel memory systems and other applied problems motivate the study of rainbow coloring of a graph, in which there is a fixed template T (or a family of templates), and one seeks to color the vertices of an input graph G with as few colors as possible, so that each copy of T in G is rainbow colored, i.e., has no two vertices the same color. In the above example, the data structure is modeled as the host graph G, and the specified subgraph as the template T. We call such coloring a template-driven rainbow coloring (or TR-coloring). For large data sets, it is also important to ensure that no memory bank (color) is overloaded, i.e., the coloring is as balanced as possible. Additionally, for fast access to data, it is desirable to quickly determine the address of a memory bank storing a data item. For arbitrary topology of G and T, finding an optimal and balanced TR-coloring is a challenging problem. This paper focuses on rainbow coloring of proper interval graphs (as hosts) for cycle templates. In particular, we present an O(k⋅|V|+|E|) time algorithm to find a TR-coloring of a proper interval graph G with respect to k-length cycle template, Ck. Our algorithm produces a coloring that is (i) optimal, i.e., it uses minimum possible number of colors in any TR-coloring; (ii) balanced, i.e, the vertices are evenly distributed among the different color classes; and (iii) explicit, i.e., the color assigned to a vertex can be computed by a closed form formula in constant time.

In this brief paper we survey some of the recent development related to the notion of homomorphism. We briefly survey the many faceted development of the homomorphism concept in the last 50 years with emphasizes on recent developments.

There has been much progress recently on establishing the complexity of various homomorphism-related computational problems. This chapter is intended to serve as a gentle introduction to graph homomorphisms in general and the complexity of homomorphism problems in particular. We also present a survey of recent progress on the complexity problems, and mention some of our favourite open questions in this area.

Barnette identified two interesting classes of cubic polyhedral graphs for which he conjectured the existence of a Hamiltonian cycle. We examine these classes from the point of view of distance-two colourings. A distance-two r-colouring of a graph G is an assignment of r colours to the vertices of G so that any two vertices at distance at most two have different colours. A cubic graph obviously needs at least four colours, and the distance-two four-colouring problem for cubic planar graphs is known to be NP-complete. We prove the problem remains NP-complete for tri-connected bipartite cubic planar graphs, which we call type-one Barnette graphs, since they are the first class identified by Barnette. By contrast, we show that the problem is polynomial for cubic plane graphs with face sizes 3,4,5, or 6, which we call type-two Barnette graphs, because of their relation to Barnette's second conjecture. In fact, the colourable instances that are bipartite can be fully described, and those that are non-bipartite can be characterized by their face sizes. We have similar results for quartic plane graphs: the analogue of type-two Barnette graphs are graphs with face sizes 3 or 4. For this class, the corresponding distance-two five-colouring problem is also polynomial; in fact, we can again fully describe all colourable instances — there are exactly two such graphs. It has recently been proved that every planar subcubic graph can be distance-two coloured with at most seven colours, and conjectured that six colours suffice if the planar subcubic graph can be drawn without faces of size five. We consider a weaker version of the conjecture, stating that six colours suffice for a bipartite cubic planar graph, i.e., a cubic plane graph with all faces even. We prove this conjecture in the case when all faces have sizes divisible by four, and in another more general case, when the faces are coloured in three colours, of which two colours consist of faces with sizes divisible by four.

We unify two popular graph classes, strongly chordal graphs and chordal bigraphs, by introducing an umbrella class that contains both classes and maintains their essential properties. This is done by allowing loops at vertices. Considering loops often has little impact on a class of graphs; it however makes a big difference in this case. We call the new class \itstrongly chordal graphs with possible loops. When all vertices have loops, we recover the usual strongly chordal graphs; when all vertices are loopless, we obtain the usual chordal bigraphs. Moreover, there is a surprizing wealth of graphs in the new class that have loops at some vertices and not at others. These graphs also admit the elegant algorithms previously only applied in the extreme two cases. Formulated in the language of adjacency matrices, we study the class of symmetric 0, 1 matrices that admit a simultaneous row and column permutation avoiding the $\Gamma$ matrix $[ \begin{smallmatrix} 1 \ 1 \\ 1 \ 0 \end{smallmatrix}]$. We give ordering characterizations, matrix characterizations, and forbidden subgraph characterizations of the new class, and illustrate its usefulness by solving the minimum domination problem in this general context. This implies solutions of both the minimum dominating set in strongly chordal graphs and the minimum total dominating set in chordal bigraphs.

We propose bipartite analogues of comparability and cocomparability graphs. Surprisingly, the two classes coincide. We call these bipartite graphs cocomparability bigraphs. We characterize cocomparability bigraphs in terms of vertex orderings, forbidden substructures, and orientations of their complements. In particular, we prove that cocomparability bigraphs are precisely those bipartite graphs that do not have edge-asteroids; this is analogous to Gallai's structural characterization of cocomparability graphs by the absence of (vertex-) asteroids. Our characterizations imply a robust polynomial-time recognition algorithm for the class of cocomparability bigraphs. Finally, we also discuss a natural relation of cocomparability bigraphs to interval containment bigraphs, resembling a well-known relation of cocomparability graphs to interval graphs.

Weighted Szeged index is a recently introduced extension of the well-known Szeged index. In this paper, we present a new tool to analyze and characterize minimum weighted Szeged index trees. We exhibit the best trees with up to 81 vertices and use this information, together with our results, to propose various conjectures on the structure of minimum weighted Szeged index trees.

We consider a variation of arboricity, where a graph is partitioned into p forests and q independent sets. These problems are NP-complete in general, but polynomial-time solvable in the class of cographs; in fact, for each p and q there are only finitely many minimal non-partitionable cographs. In previous investigations it was revealed that when $$p=0$$ or $$p=1$$, these minimal non-partitionable cographs can be uniformly described as one family of obstructions valid for all values of q. We investigate the next case, when $$p=2$$; we provide the complete family of minimal obstructions for $$p=2, q=1$$, and find that they include more than just the natural extensions of the previously described obstructions for $$p=2, q=0$$. Thus a uniform description for all q seems unlikely already in the case $$p=2$$. Our result gives a concrete forbidden induced subgraph characterization of cographs that can be partitioned into two forests and one independent set. Since our proof is algorithmic, we can apply our characterization to complement the recognition algorithm for partitionable cographs by an algorithm to certify non-partitionable cographs by finding a forbidden induced subgraph.