Convex Hulls and Simple Colourings in Directed and 2-edge-Coloured Graphs

04/18/2020 ∙ by Christopher Duffy, et al. ∙ University of Saskatchewan 0

An oriented graph (2-edge-coloured graph) is complete convex when the convex hull of every arc (edge) is the entirety of the vertex set. Here we show the problem of bounding the oriented (2-edge-coloured) chromatic number of the family of planar graphs can be restricted to bounding this parameter for the family of planar complete-convex oriented (2-edge-coloured) graphs. We fully classify complete-convex oriented and 2-edge-coloured graphs with tree-width 2. Further we show that it is NP-complete to decide if a graph is the underlying graph of a complete-convex oriented or 2-edge-coloured graph even when restricted to inputs with tree-width 4.

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1. Introduction and Preliminary Notions

The main matter of this paper concerns homomorphisms of two objects that arise from graphs with no parallel edges – oriented graphs and -edge-coloured graphs. An oriented graph, , arises from a graph by assigning to each edge a direction to form an arc. Alternately an oriented graph is an anti-symmetric digraph. A -edge-coloured graph arises from a graph by assigning a colour, red or blue, to each edge. Alternately, a -edge-coloured graph is a pair where is a graph and . In each case we refer to as the underlying graph. We assume graphs are irreflexive. We further assume graphs are non-trivial – that is, they have at least two vertices. For other notation and definitions not defined herein we refer to [4].

For an oriented graph with we say that is a -dipath and that is the centre of the -dipath. For an -edge-coloured graph with we say that is a -path, and that is the centre of the -path. When we say the -path is alternating. Otherwise we say the -path is monochromatic.

Let and be oriented graphs. We say there is a homomorphism of to when there exists so that for each we have . When there is a homomorphism of to we write . We say that is a homomorphism or a -colouring and we write .

Similarly we define the notion of homomorphism for -edge-coloured graphs. Let and be -edge-coloured graphs. We say there is a homomorphism of to when there exists so that for each we have and . When there is a homomorphism of to we write . We say that is a homomorphism or a -colouring and we write .

As graph colouring can be defined by way of graph homomorphism, we analogously define colouring for oriented and -edge-coloured graphs.

Let be an oriented graph. The chromatic number of is the least integer that for some tournament with vertices. We denote this parameter as . For a family of oriented graphs we define to be the least integer such that for each .

Let be -edge-coloured graph. The chromatic number of is the least integer that for some -edge-coloured complete graph with vertices. We denote this parameter as . For a family of -edge-coloured graphs we define to be the least integer such that for each .

The respective definitions of chromatic number by way of homomorphism are equivalent to the following definitions of chromatic number.

Let be an oriented graph. The chromatic number of is the least integer such that there exists so that

  1. for each ;

  2. for each ; and

  3. for if , then .

Let be a -edge-coloured graph. The chromatic number of is the least integer such that there exists so that

  1. for each ;

  2. for each with ; and

  3. for with if , then .

The interested reader can find more on these notions of chromatic number in [10] and in [3].

A major landmark in the study of colourings of oriented and -edge-coloured graphs are upper bounds on the chromatic number of planar graphs.

Theorem 1.1.

[6] Let be a planar graph. For any orientation we have .

Theorem 1.2.

[1] Let be a planar graph. For any -edge-coloured graph we have .

Not believed to be tight, these bounds have stood unimproved for more than twenty years. Let be the family of orientations of planar graphs. Let be the family of -edge-coloured planar graphs. A main contribution of this article is the description of non-trivial subclasses and so that and . This then reduces the task of bounding and to bounding and . We arrive at these subclasses through the following relaxation of the definitions of oriented and -edge-coloured colouring.

Let be an oriented graph. The simple chromatic number of is the least integer such that there exists so that

  1. there exists with ;

  2. for each , if then ; and

  3. for , if and , then .

Let be a -edge-coloured graph. The simple chromatic number of is the least integer such that there exists so that

  1. there exists with ;

  2. for each with , if then ; and

  3. for with , if and , then .

We use to denote the simple chromatic number of an oriented graph or a -edge-coloured graph. Intuitively, one may understand simple colourings of oriented and -edge-coloured graphs to be colourings (as defined above) in which adjacent vertices may get the same colour and in which at least two colours are used. Simple colourings were first introduced in [9] for oriented graphs. The definition of simple colouring may be re-framed in terms of homomorphism with reflexive targets. See [2] and [8] for details.

We notice that every colouring of an oriented or -edge-coloured graph is also simple colouring. Thus for any oriented or -edge-coloured graph we have . In this work we study oriented or -edge-coloured graphs for which . This work is particularly relevant in light of the following two results.

Theorem 1.3.

[8] For the family of orientations of planar graphs we have

Theorem 1.4.

[2] For the family of -edge-coloured of planar graphs we have

Let be an oriented graph and consider . The convex hull of , denoted , is the smallest subset of so that and for each , vertex is not the centre vertex of a -dipath whose ends are contained in . When is an induced subgraph of we use to denote .

Similarly, let be a -edge-coloured graph and consider . The convex hull of , denoted , is the smallest subset of so that so that and for each , vertex is not the centre vertex of an alternating -path whose ends are contained in . When is an induced subgraph of we use to denote .

Equivalently one may define the convex hull of a set of vertices as follows. Let be an oriented or -edge-coloured graph and consider . Define the sequence so that

  • ;

  • for each let where is the set of vertices that are the centre of a -dipath or alternating path whose ends are in .

Since and is finite, there exists a least integer so that . It is easily checked that .

With this definition it is clear that if , then . Also notice that if an edge is not contained in an induced copy of , then for every oriented graph or -edge-coloured graph whose underlying graph is . By way of example, consider the -edge-coloured graph in Figure 1. Observe that identifying the vertices of into a single vertex yields a -edge-coloured graph.

Figure 1. The convex hull of and

Smolíková introduces this notion of convex hull for oriented graphs in [9]. This notion is extended to -edge-coloured graphs in [2]. In both cases the following key observation arises.

Lemma 1.5.

Let be an oriented (-edge-coloured) graph. Let be a simple colouring of . If for (), then for each .

Let be an oriented (-edge-coloured) graph. We say that is complete convex when for each () we have . Notice that the -edge-coloured graph in Figure 1 is not complete convex. See Figure 2 for an example of a complete-convex oriented graph and complete-convex -edge-coloured graph. In this case we notice that the wheel on vertices is the underlying graph of both a complete-convex oriented graph and a complete-convex -edge-coloured graph. In Section 2 we will see that this is not the case for all graphs.

Lemma 1.5 implies that any simple colouring of a complete-convex oriented (-edge-coloured) graph is necessarily proper (recall that at least two colours must be used in a simple colouring). Thus any simple colouring of a complete convex oriented (-edge-coloured) graph is in fact a colouring.

Figure 2. A complete-convex oriented graph and a complete-convex -edge-coloured graph
Theorem 1.6.

[2] Let be an oriented or -edge-coloured graph. If is complete convex then .

The remainder of this work is a study of complete-convex oriented and -edge-coloured graphs. In Section 2 we study the structure of complete-convex oriented and -edge-coloured graphs. In doing so we fully classify complete-convex oriented and -edge-coloured graphs with tree-width . This classification implies the existence of graphs for which complete-convex orientations may be found but complete-convex -edge colourings may not. In Section 3, we show to be NP-complete the problem of deciding if a graph is the underlying graph of some complete-convex -edge-coloured or oriented graph. In Section 4 we find a relationship between the simple chromatic number and chromatic number for minor-closed families of oriented and -edge-coloured graphs. We use these results to compute the simple chromatic number of some well-studied families of minor-closed graphs.

2. The Structure of Complete Convex Oriented and -edge Coloured Graphs

We begin with two näive, but useful observations.

Lemma 2.1.

Let and be oriented graphs. If and are each complete convex, then any oriented graph formed by identifying any arc of with any arc of is complete convex.

Proof.

Let and be complete-convex oriented graphs. Consider and . Let be the oriented graph produced by identifying and . Let . We observe . Since and are each complete convex, for any we have . Therefore . Therefore is complete convex. ∎

A similar result holds for -edge-coloured graphs.

Lemma 2.2.

Let and be -edge-coloured graphs. If and are complete convex, then any -edge-coloured graph formed from by identifying a red (blue) edge in and a red (blue) edge in is complete convex.

With an eye towards graphs with tree-width , we continue our study with an observation of vertices of degree in complete-convex oriented graphs.

Lemma 2.3.

Let be a graph and let be a vertex of degree in . If is a complete-convex oriented graph, then is contained in a directed -cycle in .

Proof.

Let be a complete-convex oriented graph and be a vertex with degree in . Let and be the neighbours of in . Since is complete convex, each vertex of must be the centre vertex of a -dipath in . Without loss of generality, assume . If , then . Thus the subgraph induced by is a directed -cycle. ∎

Lemma 2.4.

Let be an oriented graph with a vertex of degree . If is complete convex, then is complete convex.

Proof.

Assume is complete convex. Consider the oriented graph . If is not complete convex, then there exists so that . Let . Since is complete convex, it must be that . Since , the set is non-empty. Since , there exists so that is the centre vertex of a -dipath whose ends are in . This -dipath does not exist in , thus and so .

Consider now . Again, since is non-empty and , there exists so that is the centre vertex of a -dipath whose ends are in . Since , vertex cannot be an end of this -dipath. Let be the ends of this -dipath. Notice that . Thus is a -dipath whose ends are in and whose centre vertex is in . This contradicts that . Therefore is complete convex. ∎

Lemmas 2.3 and 2.4 allow us to fully classify those graphs with tree-width that admit a complete-convex orientation.

Theorem 2.5.

Let be a graph with tree-width . An oriented graph is complete convex if and only if is a -tree and every induced copy of in is a directed -cycle in .

Proof.

Assume is a -tree. Let be an orientation of in which every induced is oriented as a directed -cycle. We show is complete convex by induction on , the number of vertices of . Notice that the directed -cycle is complete convex. Assume that has vertices. Let be a vertex of degree . By induction is complete convex. It follows from Lemma 2.1 that is complete convex.

Consider now a graph with tree-width so that is complete-convex. Since has tree-width , is a spanning subgraph of a -tree, . Since is a -tree, there is an ordering of its vertices: so that and for each vertex has degree and is contained in a copy of in .

If is a proper subgraph of , then there is a largest index such that has degree less than in . By Lemma 2.4, the oriented graph is complete convex. The only complete-convex oriented graph with a vertex of degree is . Therefore is not complete convex. This is a contradiction. Thus is a -tree.

It remains to show that every induced copy of is a directed -cycle in . Assume otherwise. Thus there exists a greatest index so that the copy of in that contains is not a directed -cycle in . By Lemma 2.3, is complete convex. This contradicts the statement of Lemma 2.3 as has degree in . ∎

Lemma 2.4 gives a method to add/remove vertices of degree to a complete-convex oriented graph so that the resulting oriented graph is complete convex: Let be a complete-convex oriented graph with vertices. For any arc one may add a new vertex and arcs to form a complete-convex oriented graph with vertices. We now describe a method of adding/removing arcs to a complete-convex oriented graph so that the resulting oriented graph is complete convex.

Let be oriented graph and let . Denote by the oriented graph formed from by reversing the orientation of . Denote by the oriented graph formed from by removing .

Theorem 2.6.

Let be an oriented graph and let . If and are complete convex, then is complete convex.

Proof.

Let be an oriented graph and let so that and are complete convex. If is not complete convex then there exists so that . Let . If or , then . Thus, without loss of generality, assume and . Since , but , there exists so that . Similarly, since , but , there exists so that . Therefore is a -dipath so that and . Therefore , a contradiction. ∎

Theorem 2.7.

Let be a complete-convex oriented graph. If and and are the ends of a -dipath in , then each of and is complete convex.

Proof.

Assume is a complete-convex oriented graph. Consider so that is a -dipath in for some . Notice . Therefore . Since , it follows that . Therefore . And so it follows that is complete convex.

A similar argument shows is complete convex. ∎

For -edge-coloured graphs a different picture emerges when we examine vertices minimum degree.

Lemma 2.8.

A complete-convex -edge-coloured graph is either a monochromatic copy of or has minimum degree .

Proof.

Consider . Let be a complete-convex -edge-coloured graph. Let be a vertex of minimum degree in .

If has a single neighbour, say , in , then . Assume has two neighbors, say and , in . Since is complete convex, must be the centre vertex of an alternating -path and and must be adjacent. Without loss of generality, let and . However we notice that . ∎

Theorem 2.9.

No graph with tree-width is the graph underlying a complete-convex -edge-coloured graph.

Theorems analogous to Theorems 2.6 and 2.7 hold for -edge-coloured graphs. For the analogue of Theorem 2.6 the notion of reversing the orientation of an arc is replaced with the notion of changing the colour of an edge. For the analogue of Theorem 2.7 we replace -dipath with alternating -path.

3. Complexity

We study the following decision problems.

Problem: COMPLETE CONVEX 2EC
Instance: A graph .
Decide: Does there exist a complete-convex -edge-coloured graph ?

Problem: COMPLETE CONVEX ORIENT
Instance: A graph .
Decide: Does there exist a complete-convex orientation of ?

Theorems 2.5 and 2.9 imply that there are YES instances of COMPLETE CONVEX ORIENT that are NO instances of COMPLETE CONVEX 2EC. In [2] it is shown that there are NO instances of COMPLETE CONVEX ORIENT that are YES instances of COMPLETE CONVEX 2EC.

We study these problems through a reduction from monotone not-all-equal satisfiability.

Problem: MONOTONE NAE3SAT
Instance: A monotone boolean formula in conjunctive normal form with three variables in each clause.
Decide: Does there exist a not-all-equal satisfying assignment for the elements of ?

Without loss of generality, we assume that in an instance of MONOTONE NAE3SAT no partition of induces a partition of .

Theorem 3.1.

[7] The decision problem MONOTONE NAE3SAT is NP-complete.

Let be an instance of MONOTONE NAE3SAT. We construct the graph as follows. For each we construct as shown in Figure 3.

Figure 3. The clause graph, , for .

We call these refer to these graphs as clause graphs. We construct from the set of clause graphs for as follows:

  • Identify all vertices labelled .

  • For each identify all vertices labelled .

We note that given , the graph can be constructed in polynomial time.

We show that there exists complete-convex -edge-coloured graph if and only if is not-all-equal satisfiable. For each , the colour of the edge will represent the assignment for in a not-all-equal satisfying assignment for . We begin with three technical lemmas.

Lemma 3.2.

Consider a complete-convex -edge-coloured graph . For every and every so that is a literal of we have .

Proof.

Assume is complete convex. Without loss of generality, assume . There is only one -path with ends and – this is the path . Since and is complete convex, it follows that . By observing that there is only one -path with ends and it similarly follows that . ∎

Lemma 3.3.

Consider complete convex . For every we have

Proof.

We proceed by contradiction. Without loss of generality, assume . There are two -paths with ends and – the path and the path . Since and is complete convex it follows that . Without loss of generality, assume and . There are two -paths with ends and – the path and the path . Since and is complete convex it follows that . Therefore . However now we notice . This is a contradiction as is complete convex. ∎

For a clause and a variable that appears as a literal of , denote by the subgraph induced by . For a clause denote by the subgraph induced by .

Lemma 3.4.

Consider . If each of the -edge-coloured subgraphs of the form and is complete convex, then is complete convex.

Proof.

We proceed by induction on the number of clauses in . If has a single clause, then , where is the lone clause in . By Lemma 2.2, it follows that is complete convex.

Consider now and with . Without loss of generality, appears in some other clause of . (Recall that we may assume that no partition of induces a partition of ) Let be the instance of NAE3SAT formed by removing from . By induction is complete convex. The result now follows from Lemma 2.2. ∎

For a clause that contains literal , the colour of the edge will correspond to the value of the literal in the clause . Lemma 3.2 implies that for a fixed literal, all such edges have the same colour. Lemma 3.3 implies that for a fixed clause the literals are not all equal.

Lemma 3.5.

Given , an instance of MONOTONE NAE3SAT, there exists a complete-convex -edge-coloured graph if and only if is not-all-equal satisfiable.

Proof.

Let be an instance of MONOTONE NAE3SAT. Consider , a -edge-coloured complete convex graph. Construct so

  • when ; and

  • when .

We claim is not-all-equal satisfying for . Consider with . By Lemma 3.3 we have . By Lemma 3.2 we have . Therefore . Therefore is not-all-equal satisfying for .

Consider now so that is not-all-equal satisfying for . We construct as follows. For all so that is a literal of and let

  • .

For all so that is a literal of and let

  • .

For with so that , let

  • ; and

  • .

For with so that , let

  • ; and

  • .

(See Figure 4 for the case , .)

Figure 4. A -edge-coloured clause graph for , when and .

By inspection, each of the -edge-coloured subgraphs of the form and is complete convex (See Figures 5 and 6) Thus by Lemma 3.4 we have that is complete convex. ∎

Figure 5. for the case and
Figure 6. when (left) and (right)
Theorem 3.6.

The decision problem COMPLETE CONVEX 2EC is NP-complete.

Proof.

Verifying that a -edge-coloured graph is complete convex is Polynomial. Therefore COMPLETE CONVEX 2EC is in NP. Given , the graph can constructed in polynomial time. By Lemma 3.5 and Theorem 3.1, COMPLETE CONVEX 2EC is NP-complete. ∎

We turn now to oriented graphs. We proceed similarly as in the argument for -edge-coloured graphs. Given we construct as above. We form from by adding a vertex and edges for each clause . We show there exists a complete convex orientation of if and only if is not-all-equal satisfiable. For each the orientation of the edge will represent the assignment for in a not-all-equal satisfying assignment for . We begin with three technical lemmas in the spirit of Lemmas 3.2, 3.3 and 3.4. The proofs of these lemmas follow similarly to those of Lemmas 3.2, 3.3 and 3.4 and are thus omitted.

Lemma 3.7.

Consider a complete convex orientation of . For every and every so that is a literal of we have that is not a -dipath.

Lemma 3.8.

Consider a complete convex orientation of . For every we have that is not a source or sink in the subgraph induced by

For a clause and a variable that appears as a literal of , denote by the subgraph induced by . For a clause denote by the subgraph induced by .

Lemma 3.9.

Consider . If each of the oriented subgraphs of the form and is complete convex, then is complete convex.

Lemma 3.10.

Given , an instance of MONOTONE NAE3SAT, there exists a complete convex orientation if and only if is not-all-equal satisfiable.

Proof.

Let be an instance of MONOTONE NAE3SAT

Consider , an orientation of Construct so

  • when .

  • when .

We claim is not-all-equal satisfying for . Consider with . By Lemma 3.8 we have that is not a source or a sink in the subgraph induced by . Therefore . By Lemma 3.7 we have that the edge is oriented towards if and only if the edge is oriented towards . Similarly, the edge is oriented towards if and only if the edge is oriented towards and the edge is oriented towards if and only if the edge is oriented towards . Therefore is not-all-equal satisfying for .

Consider now so that is not-all-equal satisfying for . For all so that is a literal of and

  • orient the edges and to have their tails at ;

  • orient the edges and to have their tails at