Clique-width and Well-Quasi-Ordering of Triangle-Free Graph Classes

1 Introduction

A graph class $G$ is well-quasi-ordered by some containment relation if for any infinite sequence $G_{0}, G_{1}, \dots$ of graphs in $G$ , there is a pair $i, j$ with $i < j$ such that $G_{i}$ is contained in $G_{j}$ . A graph class $G$ has bounded clique-width if there exists a constant $c$ such that every graph in $G$ has clique-width at most $c$ . Both being well-quasi-ordered and having bounded clique-width are highly desirable properties of graph classes in the areas of discrete mathematics and theoretical computer science. To illustrate this, let us mention the seminal project of Robertson and Seymour on graph minors that culminated in 2004 in the proof of Wagner’s conjecture, which states that the set of all finite graphs is well-quasi-ordered by the minor relation. As an algorithmic consequence, given a minor-closed graph class, it is possible to test in cubic time whether a given graph belongs to this class. The algorithmic importance of having bounded clique-width follows from the fact that many well-known NP-hard problems, such as Graph Colouring and Hamilton Cycle, become polynomial-time solvable for graph classes of bounded clique-width (this follows from combining results from several papers [6, 16, 19, 27] with a result of Oum and Seymour [25]).

Courcelle [5] proved that the class of graphs obtained from graphs of clique-width $3$ via one or more edge contractions has unbounded clique-width. Hence the clique-width of a graph can be much smaller than the clique-width of its minors. On the other hand, the clique-width of a graph is at least the clique-width of any of its induced subgraphs (see, for example, [7]). We therefore focus on hereditary classes, that is, on graph classes that are closed under taking induced subgraphs. It is readily seen that a class of graphs is hereditary if and only if it can be characterised by a unique set $F$ of minimal forbidden induced subgraphs. Our underlying research goal is to increase our understanding of the relation between well-quasi-orders and clique-width of hereditary graph classes.

As a start, we note that the hereditary class of graphs of degree at most $2$ is not well-quasi-ordered by the induced subgraph relation, as it contains the class of cycles, which form an infinite antichain. As every graph of degree at most $2$ has clique-width at most $4$ , having bounded clique-width does not imply being well-quasi-ordered by the induced subgraph relation. In 2010, Daligault, Rao and Thomassé [13] asked about the reverse implication: does every hereditary graph class that is well-quasi-ordered by the induced subgraph relation have bounded clique-width? In 2015, Lozin, Razgon and Zamaraev [24] gave a negative answer. As the set $F$ of minimal forbidden induced subgraphs in their counter-example is infinite, the question of Daligault, Rao and Thomassé [13] remains open for finitely defined hereditary graph classes, that is, hereditary graph classes for which $F$ is finite.

Conjecture 1 ([24])

If a finitely defined hereditary class of graphs $G$ is well-quasi-ordered by the induced subgraph relation, then $G$ has bounded clique-width.

If Conjecture 1 is true, then for finitely defined hereditary graph classes the aforementioned algorithmic consequences of having bounded clique-width also hold for the property of being well-quasi-ordered by the induced subgraph relation. A hereditary graph class defined by a single forbidden induced subgraph $H$ is well-quasi-ordered by the induced subgraph relation if and only if it has bounded clique-width if and only if $H$ is an induced subgraph of $P_{4}$ (see, for instance, [12, 14, 21]). Hence Conjecture 1 holds when $F$ has size $1$ . We consider the case when $F$ has size $2$ , say $F = {H_{1}, H_{2}}$ . Such graph classes are said to be bigenic or $(H_{1}, H_{2})$ -free graph classes. In this case Conjecture 1 is also known to be true except for two stubborn open cases, namely $(H_{1}, H_{2}) = (K_{3}, P_{2} + P_{4})$ and $(H_{1}, H_{2}) = (¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ P_{1} + P_{4}, P_{2} + P_{3})$ ; see [10].

Our Results. In Section 4, we prove that Conjecture 1 holds for the class of $(K_{3}, P_{2} + P_{4})$ -free graphs by showing that the class of $(K_{3}, P_{2} + P_{4})$ -free graphs has bounded clique-width and is well-quasi-ordered by the induced subgraph relation. We do this by using a general technique explained in Section 3. This technique is based on extending (a labelled version of) well-quasi-orderability or boundedness of clique-width of the bipartite graphs in a hereditary graph class $X$ to a special subclass of $3$ -partite graphs in $X$ . The crucial property of these $3$ -partite graphs is that no three vertices from the three different partition classes form a clique or independent set. We say that such $3$ -partite graphs curious. A more restricted version of this concept was used to prove that $(K_{3}, P_{1} + P_{5})$ -free graphs have bounded clique-width [8]. In Section 4 we show how to generalise results for curious $(K_{3}, P_{2} + P_{4})$ -free graphs to the whole class of $(K_{3}, P_{2} + P_{4})$ -free graphs. In the same section we also show how to apply our technique to prove that the class of $(K_{3}, P_{1} + P_{5})$ -free graphs is well-quasi-ordered (it was already known [8] that the class of $(K_{3}, P_{1} + P_{5})$ -free graphs has bounded clique-width). We note that our results also imply that the class of $(K_{3}, P_{1} + 2 P_{2})$ -free graphs is well-quasi-ordered, which was not previously known (see [2, 21]). See also Figure 1 for pictures of the forbidden induced subgraphs mentioned in this paragraph.


$K_{3}$	$P_{1} + 2 P_{2}$	$P_{1} + P_{5}$	$P_{2} + P_{4}$

Figure 1: The forbidden induced subgraphs considered in our results.

Dichotomies. Previously, well-quasi-orderability was known for $(K_{3}, P_{6})$ -free graphs [2], $(P_{2} + P_{4})$ -free bipartite graphs [20] and $(P_{1} + P_{5})$ -free bipartite graphs [20]. It has also been shown that $H$ -free bipartite graphs are not well-quasi-ordered if $H$ contains an induced $3 P_{1} + P_{2}$ [21], $3 P_{2}$ [15] or $2 P_{3}$ [20]. Moreover, for every $s \geq 1$ , the class of $(K_{3}, s P_{1})$ -free graphs is finite due to Ramsey’s Theorem [26]. The above results lead to the following known dichotomy for $H$ -free bipartite graphs.

Theorem 1.1

Let $H$ be a graph. The class of $H$ -free bipartite graphs is well-quasi-ordered by the induced subgraph relation if and only if $H = s P_{1}$ for some $s \geq 1$ or $H$ is an induced subgraph of $P_{1} + P_{5}$ , $P_{2} + P_{4}$ or $P_{6}$ .

Now combining the aforementioned known results for $(K_{3}, H)$ -free graphs and $H$ -free bipartite graphs with our new results yields the following new dichotomy for $H$ -free triangle-free graphs, which is exactly the same as the one in Theorem 1.1.

Theorem 1.2

Let $H$ be a graph. The class of $(K_{3}, H)$ -free graphs is well-quasi-ordered by the induced subgraph relation if and only if $H = s P_{1}$ for some $s \geq 1$ or $H$ is an induced subgraph of $P_{1} + P_{5}$ , $P_{2} + P_{4}$ , or $P_{6}$ .

Besides our technique based on curious graphs, we also expect that Theorem 1.2 will itself be a useful ingredient for showing results for other graph classes, just as Theorem 1.1 has already proven to be useful (see e.g. [20]).

For clique-width the following dichotomy is known for $H$ -free bipartite graphs.

Theorem 1.3 ([11])

Let $H$ be a graph. The class of $H$ -free bipartite graphs has bounded clique-width if and only if $H = s P_{1}$ for some $s \geq 1$ or $H$ is an induced subgraph of $K_{1, 3} + 3 P_{1}$ , $K_{1, 3} + P_{2}$ , $P_{1} + S_{1, 1, 3}$ or $S_{1, 2, 3}$ .

It would be interesting to determine whether $(K_{3}, H)$ -free graphs allow the same dichotomy with respect to the boundedness of their clique-width. The evidence so far is affirmative, but in order to answer this question two remaining cases need to be solved, namely $(H_{1}, H_{2}) = (K_{3}, P_{1} + S_{1, 1, 3})$ and $(H_{1}, H_{2}) = (K_{3}, S_{1, 2, 3})$ ; see Section 2 for the definition of the graph $S_{h, i, j}$ . Both cases turn out to be highly non-trivial; in particular, the class of $(K_{3}, P_{1} + S_{1, 1, 3})$ -free graphs contains the class of $(K_{3}, P_{1} + P_{5})$ -free graphs, and the class of $(K_{3}, S_{1, 2, 3})$ -free graphs contains both the classes of $(K_{3}, P_{1} + P_{5})$ -free and $(K_{3}, P_{2} + P_{4})$ -free graphs.

In Section 5 we give state-of-the-art summaries for well-quasi-orderability and boundedness of clique-width of bigenic graph classes (which include our new results), together with an overview of the missing cases for both problems (which include the missing cases mentioned in this section).

2 Preliminaries

Throughout the paper, we consider only finite, undirected graphs without multiple edges or self-loops. Below, we define further graph terminology.

The disjoint union $(V (G) \cup V (H), E (G) \cup E (H))$ of two vertex-disjoint graphs $G$ and $H$ is denoted by $G + H$ and the disjoint union of $r$ copies of a graph $G$ is denoted by $r G$ . The complement $¯ ¯¯ ¯ G$ of a graph $G$ has vertex set $V (¯ ¯¯ ¯ G) = V (G)$ and an edge between two distinct vertices $u, v$ if and only if $u v \notin E (G)$ . For a subset $S \subseteq V (G)$ , we let $G [S]$ denote the subgraph of $G$ induced by $S$ , which has vertex set $S$ and edge set ${u v | u, v \in S, u v \in E (G)}$ . If $S = {s_{1}, \dots, s_{r}}$ then, to simplify notation, we may also write $G [s_{1}, \dots, s_{r}]$ instead of $G [{s_{1}, \dots, s_{r}}]$ . We use $G ∖ S$ to denote the graph obtained from $G$ by deleting every vertex in $S$ , that is, $G ∖ S = G [V (G) ∖ S]$ . We write $G^{'} \subseteq_{i} G$ to indicate that $G^{'}$ is an induced subgraph of $G$ .

The graphs $C_{r}$ , $K_{r}$ , $K_{1, r - 1}$ and $P_{r}$ denote the cycle, complete graph, star and path on $r$ vertices, respectively. The graphs $K_{3}$ and $K_{1, 3}$ are also called the triangle and claw, respectively. A graph $G$ is a linear forest if every component of $G$ is a path (on at least one vertex). The graph $S_{h, i, j}$ , for $1 \leq h \leq i \leq j$ , denotes the subdivided claw, that is, the tree that has only one vertex $x$ of degree $3$ and exactly three leaves, which are of distance $h$ , $i$ and $j$ from $x$ , respectively. Observe that $S_{1, 1, 1} = K_{1, 3}$ . We let $S$ denote the class of graphs, each connected component of which is either a subdivided claw or a path.

For a set of graphs ${H_{1}, \dots, H_{p}}$ , a graph $G$ is $(H_{1}, \dots, H_{p})$ -free if it has no induced subgraph isomorphic to a graph in ${H_{1}, \dots, H_{p}}$ ; if $p = 1$ , we may write $H_{1}$ -free instead of $(H_{1})$ -free. A graph is $k$ -partite if its vertex can be partitioned into $k$ (possibly empty) independent sets; $2$ -partite graphs are also known as bipartite graphs. The biclique or complete bipartite graph $K_{r, s}$ is the bipartite graph with sets in the partition of size $r$ and $s$ respectively, such that every vertex in one set is adjacent to every vertex in the other set. For a graph $G = (V, E)$ , the set $N (u) = {v \in V | u v \in E}$ denotes the neighbourhood of $u \in V$ . Let $X$ be a set of vertices in $G$ . A vertex $y \in V ∖ X$ is complete to $X$ if it is adjacent to every vertex of $X$ and anti-complete to $X$ if it is non-adjacent to every vertex of $X$ . A set of vertices $Y \subseteq V ∖ X$ is complete (resp. anti-complete) to $X$ if every vertex in $Y$ is complete (resp. anti-complete) to $X$ . If $X$ and $Y$ are disjoint sets of vertices in a graph, we say that the edges between these two sets form a matching if each vertex in $X$ has at most one neighbour in $Y$ and vice versa (if each vertex has exactly one such neighbour, we say that the matching is perfect). Similarly, the edges between these sets form a co-matching if each vertex in $X$ has at most one non-neighbour in $Y$ and vice versa. We say that the set $X$ dominates $Y$ if every vertex of $Y$ has at least one neighbour in $X$ . Similarly, a vertex $x$ dominates $Y$ if every vertex of $Y$ is adjacent to $x$ . A vertex $y \in V ∖ X$ distinguishes $X$ if $y$ has both a neighbour and a non-neighbour in $X$ . The set $X$ is a module of $G$ if no vertex in $V ∖ X$ distinguishes $X$ . A module $X$ is non-trivial if $1 < | X | < | V |$ , otherwise it is trivial. A graph is prime if it has only trivial modules. Two vertices are false twins if they have the same neighbourhood (note that such vertices must be non-adjacent). Clearly any prime graph on at least three vertices cannot contain a pair of false twins, as any such pair of vertices would form a non-trivial module.

We will use the following structural result.

Lemma 1 ([8])

Let $G$ be a connected $(K_{3}, C_{5}, S_{1, 2, 3})$ -free graph that does not contain a pair of false twins. Then $G$ is either bipartite or a cycle.

2.1 Clique-width

The clique-width $cw (G)$ of a graph $G$ is the minimum number of labels needed to construct $G$ by using the following four operations:

$i (v)$ : creating a new graph consisting of a single vertex $v$ with label $i$ ;
$G_{1} \oplus G_{2}$ : taking the disjoint union of two labelled graphs $G_{1}$ and $G_{2}$ ;
$η_{i, j}$ : joining each vertex with label $i$ to each vertex with label $j$ ( $i \neq j$ );
$ρ_{i \to j}$ : renaming label $i$ to $j$ .

A class of graphs $G$ has bounded clique-width if there is a constant $c$ such that the clique-width of every graph in $G$ is at most $c$ ; otherwise the clique-width is unbounded.

Let $G$ be a graph. We define the following operations. For an induced subgraph $G^{'} \subseteq_{i} G$ , the subgraph complementation operation (acting on $G$ with respect to $G^{'}$ ) replaces every edge present in $G^{'}$ by a non-edge, and vice versa. Similarly, for two disjoint vertex subsets $S$ and $T$ in $G$ , the bipartite complementation operation with respect to $S$ and $T$ acts on $G$ by replacing every edge with one end-vertex in $S$ and the other one in $T$ by a non-edge and vice versa.

We now state some useful facts about how the above operations (and some other ones) influence the clique-width of a graph. We will use these facts throughout the paper. Let $k \geq 0$ be a constant and let $γ$ be some graph operation. We say that a graph class $G^{'}$ is $(k, γ)$ -obtained from a graph class $G$ if the following two conditions hold:

every graph in $G^{'}$ is obtained from a graph in $G$ by performing $γ$ at most $k$ times, and
for every $G \in G$ there exists at least one graph in $G^{'}$ obtained from $G$ by performing $γ$ at most $k$ times.

We say that $γ$ preserves boundedness of clique-width if for any finite constant $k$ and any graph class $G$ , any graph class $G^{'}$ that is $(k, γ)$ -obtained from $G$ has bounded clique-width if and only if $G$ has bounded clique-width.

Vertex deletion preserves boundedness of clique-width [23].
Subgraph complementation preserves boundedness of clique-width [18].
Bipartite complementation preserves boundedness of clique-width [18].

Lemma 2 ([7])

Let $G$ be a graph and let $P$ be the set of all induced subgraphs of $G$ that are prime. Then $cw (G) = {max}_{H \in P} cw (H)$ .

2.2 Well-quasi-orderability

A quasi order $\leq$ on a set $X$ is a reflexive, transitive binary relation. Two elements $x, y \in X$ in this quasi-order are comparable if $x \leq y$ or $y \leq x$ , otherwise they are incomparable. A set of elements in a quasi-order is a chain if every pair of elements is comparable and it is an antichain if every pair of elements is incomparable. The quasi-order $\leq$ is a well-quasi-order if any infinite sequence of elements $x_{1}, x_{2}, x_{3}, \dots$ in $X$ contains a pair $(x_{i}, x_{j})$ with $x_{i} \leq x_{j}$ and $i < j$ . Equivalently, a quasi-order is a well-quasi-order if and only if it has no infinite strictly decreasing sequence $x_{1} ⪈ x_{2} ⪈ x_{3} ⪈ \dots$ and no infinite antichain.

For an arbitrary set $M$ , let $M^{*}$ denote the set of finite sequences of elements of $M$ . A quasi-order $\leq$ on $M$ defines a quasi-order $\leq^{*}$ on $M^{*}$ as follows: $(a_{1}, \dots, a_{m}) \leq^{*} (b_{1}, \dots, b_{n})$ if and only if there is a sequence of integers $i_{1}, \dots, i_{m}$ with $1 \leq i_{1} < \dots < i_{m} \leq n$ such that $a_{j} \leq b_{i_{j}}$ for $j \in {1, \dots, m}$ . We call $\leq^{*}$ the subsequence relation.

Lemma 3 (Higman’s Lemma [17])

If $(M, \leq)$ is a well-quasi-order then $(M^{*}, \leq^{*})$ is a well-quasi-order.

To define the notion of labelled induced subgraphs, let us consider an arbitrary quasi-order $(W, \leq)$ . We say that $G$ is a labelled graph if each vertex $v$ of $G$ is equipped with an element $l_{G} (v) \in W$ (the label of $v$ ). Given two labelled graphs $G$ and $H$ , we say that $G$ is a labelled induced subgraph of $H$ if $G$ is isomorphic to an induced subgraph of $H$ and there is an isomorphism that maps each vertex $v$ of $G$ to a vertex $w$ of $H$ with $l_{G} (v) \leq l_{H} (w)$ . Clearly, if $(W, \leq)$ is a well-quasi-order, then a graph class $X$ cannot contain an infinite sequence of labelled graphs that is strictly-decreasing with respect to the labelled induced subgraph relation. We therefore say that a graph class $X$ is well-quasi-ordered by the labelled induced subgraph relation if it contains no infinite antichains of labelled graphs whenever $(W, \leq)$ is a well-quasi-order. Such a class is readily seen to also be well-quasi-ordered by the induced subgraph relation.

Daligault, Rao and Thomassé [13] showed that every hereditary class of graphs that is well-quasi-ordered by the labelled induced subgraph relation is defined by a finite set of forbidden induced subgraphs. Korpelainen, Lozin and Razgon [22] conjectured that if a hereditary class of graphs $G$ is defined by a finite set of forbidden induced subgraphs, then $G$ is well-quasi-ordered by the induced subgraph relation if and only if it is well-quasi-ordered by the labelled induced subgraph relation. Brignall, Engen and Vatter [4] recently found a class $G^{*}$ with 14 forbidden induced subgraphs that is a counterexample for this conjecture, that is $G^{*}$ is well-quasi-ordered by the induced subgraph relation but not by the labelled induced subgraph relation. However, so far, all known results for bigenic graph classes, including those in this paper, verify the conjecture for bigenic graph classes.

Similarly to the notion of preserving boundedness of clique-width, we say that a graph operation $γ$ preserves well-quasi-orderability by the labelled induced subgraph relation if for any finite constant $k$ and any graph class $G$ , any graph class $G^{'}$ that is $(k, γ)$ -obtained from $G$ is well-quasi-ordered by this relation if and only if $G$ is.

Lemma 4

[10] The following operations preserve well-quasi-orderability by the labelled induced subgraph relation:

Subgraph complementation,
Bipartite complementation and
Vertex deletion.

Lemma 5 ([2])

A hereditary class $X$ of graphs is well-quasi-ordered by the labelled induced subgraph relation if and only if the set of prime graphs in $X$ is. In particular, $X$ is well-quasi-ordered by the labelled induced subgraph relation if and only if the set of connected graphs in $X$ is.

Lemma 6 ([2, 20])

$(P_{7}, S_{1, 2, 3})$ -free bipartite graphs are well-quasi-ordered by the labelled induced subgraph relation.

Let $(L_{1}, \leq_{1})$ and $(L_{2}, \leq_{2})$ be well-quasi-orders. We define the Cartesian Product $(L_{1}, \leq_{1}) \times (L_{2}, \leq_{2})$ of these well-quasi-orders as the order $(L, \leq_{L})$ on the set $L := L_{1} \times L_{2}$ where $(l_{1}, l_{2}) \leq_{L} (l_{1}^{'}, l_{2}^{'})$ if and only if $l_{1} \leq_{1} l_{1}^{'}$ and $l_{2} \leq_{2} l_{2}^{'}$ . Lemma 3 implies that $(L, \leq_{L})$ is also a well-quasi-order. If $G$ has a labelling with elements of $L_{1}$ and of $L_{2}$ , say $l_{1} : V (G) \to L_{1}$ and $l_{2} : V (G) \to L_{2}$ , we can construct the combined labelling in $(L_{1}, \leq_{1}) \times (L_{2}, \leq_{2})$ that labels each vertex $v$ of $G$ with the label $(l_{1} (v), l_{2} (v))$ .

Lemma 7

Fix a well-quasi-order $(L_{1}, \leq_{1})$ that has at least one element. Let $X$ be a class of graphs. For each $G \in X$ fix a labelling $l_{G}^{1} : V (G) \to L_{1}$ . Then $X$ is well-quasi-ordered by the labelled induced subgraph relation if and only if for every well-quasi-order $(L_{2}, \leq_{2})$ and every labelling of the graphs in $X$ by this order, the combined labelling in $(L_{1}, \leq_{1}) \times (L_{2}, \leq_{2})$ obtained from these labellings also results in a well-quasi-ordered set of labelled graphs.

Proof

If $X$ is well-quasi-ordered by the labelled induced subgraph relation then by definition it is well-quasi-ordered when labelled with labels from these combined labellings obtained from these well-quasi-orders. If $X$ is not well-quasi-ordered by the labelled induced subgraph relation then there must be a well-quasi-order $(L_{2}, \leq_{2})$ and an infinite set of graphs $G_{1}, G_{2}, \dots$ whose vertices are labelled with elements of $L_{2}$ such that these graphs form an infinite labelled antichain. For each graph $G_{i}$ , replace the label $l$ on vertex $v$ by $(l_{G_{1}}^{1} (v), l)$ . The graphs are now labelled with elements of the well-quasi-order $(L_{1}, \leq_{1}) \times (L_{2}, \leq_{2})$ and result in an infinite labelled antichain of graphs labelled with such combined labellings. This completes the proof.∎

2.3 $k$ -uniform Graphs

For an integer $k \geq 1$ , a graph $G$ is $k$ -uniform if there is a symmetric square $0, 1$ matrix $K$ of order $k$ and a graph $F_{k}$ on vertices $1, 2, \dots, k$ such that $G \in P (K, F_{k})$ , where $P (K, F_{k})$ is the graph class defined as follows. Let $H$ be the disjoint union of infinitely many copies of $F_{k}$ . For $i = 1, \dots, k$ , let $V_{i}$ be the subset of $V (H)$ containing vertex $i$ from each copy of $F_{k}$ . Construct from $H$ an infinite graph $H (K)$ on the same vertex set by applying a subgraph complementation to $V_{i}$ if and only if $K (i, i) = 1$ and by applying a bipartite complementation to a pair $V_{i}, V_{j}$ if and only if $K (i, j) = 1$ . Thus, two vertices $u \in V_{i}$ and $v \in V_{j}$ are adjacent in $H (K)$ if and only if $u v \in E (H)$ and $K (i, j) = 0$ or $u v \notin E (H)$ and $K (i, j) = 1$ . Then, $P (K, F_{k})$ is the hereditary class consisting of all the finite induced subgraphs of $H (K)$ . The minimum $k$ such that $G$ is $k$ -uniform is the uniformicity of $G$ . The second of the next two lemmas follows directly from the above definitions.

The following result was proved by Korpelainen and Lozin. The class of disjoint unions of cliques is a counterexample for the reverse implication.

Lemma 8 ([21])

Any class of graphs of bounded uniformicity is well-quasi-ordered by the labelled induced subgraph relation.

The following lemma follows directly from the definition of clique-width and the definition of $k$ -uniform graphs (see also [1] for a more general result).

Lemma 9

Every $k$ -uniform graph has clique-width at most $2 k$ .

3 Partitioning $3$ -Partite Graphs

In Section 3.1, we first introduce a graph decomposition on $3$ -partite graphs. We then show how to extend results on bounded clique-width or well-quasi-orderability by the labelled induced subgraph relation from bipartite graphs in an arbitrary hereditary class of graphs to the $3$ -partite graphs in this class that are decomposable in this way. In Section 3.2, we then give sufficient conditions for a $3$ -partite graph to have such a decomposition.

3.1 The Decomposition

Let $G$ be a $3$ -partite graph given with a partition of its vertex set into three independent sets $V_{1}$ , $V_{2}$ and $V_{3}$ . Suppose that each set $V_{i}$ can be partitioned into sets $V_{i}^{0}, \dots, V_{i}^{ℓ}$ such that, taking subscripts modulo $3$ :

for $i \in {1, 2, 3}$ if $j < k$ then $V_{i}^{j}$ is complete to $V_{i + 1}^{k}$ and anti-complete to $V_{i + 2}^{k}$ .

For $i \in {0, \dots, ℓ}$ let $G^{i} = G [V_{1}^{i} \cup V_{2}^{i} \cup V_{3}^{i}]$ . We say that the graphs $G^{i}$ are the slices of $G$ . If the slices belong to some graph class $X$ , then we say that $G$ can be partitioned into slices from $X$ ; see Figure 2 for an example.

Figure 2: A

3

-partite graph partitioned into slices

G^{0}, \dots, G^{3}

isomorphic to

P_{3}

Lemma 10

If $G$ is a $3$ -partite graph that can be partitioned into slices of clique-width at most $k$ then $G$ has clique-width at most $max (3 k, 6)$ .

Proof

Since every slice $G^{j}$ of $G$ has clique-width at most $k$ , it can be constructed using the labels $1, \dots, k$ . Applying relabelling operations if necessary, we may assume that at the end of this construction, every vertex receives the label $1$ . We can modify this construction so that we use the labels $1_{1}, \dots, k_{1}, 1_{2}, \dots, k_{2}, 1_{3}, \dots, k_{3}$ instead, in such a way that at all points in the construction, for each $i \in {1, 2, 3}$ every constructed vertex in $V_{i}$ has a label in ${1_{i}, \dots, k_{i}}$ . To do this we replace:

creation operations $i (v)$ by $i_{j} (v)$ if $v \in V_{j}$ ,
relabel operations $ρ_{j \to k} ()$ by $ρ_{j_{1} \to k_{1}} (ρ_{j_{2} \to k_{2}} (ρ_{j_{3} \to k_{3}} ()))$ and
join operations $η_{j, k} ()$ by
$η_{j_{1}, k_{1}} (η_{j_{1}, k_{2}} (η_{j_{1}, k_{3}} (η_{j_{2}, k_{1}} (η_{j_{2}, k_{2}} (η_{j_{2}, k_{3}} (η_{j_{3}, k_{1}} (η_{j_{3}, k_{2}} (η_{j_{3}, k_{3}} ()))))))))$ .

This modified construction uses $3 k$ labels and at the end of it, every vertex in $V_{i}$ is labelled with label $1_{i}$ . We may do this for every slice $G^{j}$ of $G$ independently. We now show how to use these constructed slices to construct $G [V (G^{0}) \cup \dots \cup V (G^{j})]$ using six labels in such a way that every vertex in $V_{i}$ is labelled with label $1_{i}$ . We do this by induction. If $j = 0$ then $G [V (G^{0})] = G^{0}$ , so we are done. If $j > 0$ then by the induction hypothesis, we can construct $G [V (G^{0}) \cup \dots \cup V (G^{j - 1})]$ in this way. Consider the copy of $G^{j}$ constructed earlier and relabel its vertices using the operations $ρ_{1_{1} \to 2_{1}}$ , $ρ_{1_{2} \to 2_{2}}$ and $ρ_{1_{3} \to 2_{3}}$ so that in this copy of $G^{j}$ , every vertex in $V_{i}$ is labelled $2_{i}$ . Next take the disjoint union of the obtained graph with the constructed $G [V (G^{0}) \cup \dots \cup V (G^{j - 1})]$ . Then, apply join operations $η_{1_{1}, 2_{2}}$ , $η_{1_{2}, 2_{3}}$ and $η_{1_{3}, 2_{1}}$ . Finally, apply the relabelling operations $ρ_{2_{1} \to 1_{1}}$ , $ρ_{2_{2} \to 1_{2}}$ and $ρ_{2_{3} \to 1_{3}}$ . This constructs $G [V (G^{0}) \cup \dots \cup V (G^{j})]$ in such a way that every vertex in $V_{i}$ is labelled with $1_{i}$ . By induction, it follows that $G$ has clique-width at most $max (3 k, 6)$ .∎

Lemma 11

Let $X$ be a hereditary graph class containing a class $Z$ . Let $Y$ be the set of $3$ -partite graphs in $X$ that can be partitioned into slices from $Z$ . If $Z$ is well-quasi-ordered by the labelled induced subgraph relation then so is $Y$ .

Proof

For each graph $G$ in $Y$ , we may fix a partition into independent sets $(V_{1}, V_{2}, V_{3})$ with respect to which the graph can be partitioned into slices from $Z$ . Let $(L_{1}, \leq_{1})$ be the well-quasi-order with $L_{1} = {1, 2, 3}$ in which every pair of distinct elements is incomparable. By Lemma 7, we need only consider labellings of graphs in $G$ of the form $(i, l (v))$ where $v \in V_{i}$ and $l (v)$ belongs to an arbitrary well-quasi-order $L$ . Suppose $G$ can be partitioned into slices $G^{1}, \dots, G^{k}$ , with vertices labelled as in $G$ . The slices along with the labellings completely describe the edges in $G$ . Suppose $H$ is another such graph, partitioned into slices $H^{1}, \dots, H^{k}$ . If $(H^{1}, \dots, H^{ℓ})$ is smaller than $(G^{1}, \dots, G^{k})$ under the subsequence relation, then $H$ is an induced subgraph of $G$ . The result follows by Lemma 3.∎

3.2 Curious Graphs

Let $G$ be a $3$ -partite graph given together with a partition of its vertex set into three independent sets $V_{1}$ , $V_{2}$ and $V_{3}$ . An induced $K_{3}$ or $3 P_{1}$ in $G$ is rainbow if it has exactly one vertex in each set $V_{i}$ . We say that $G$ is curious with respect to the partition $(V_{1}, V_{2}, V_{3})$ if it contains no rainbow $K_{3}$ or $3 P_{1}$ when its vertex set is partitioned in this way. We say that $G$ is curious if there is a partition $(V_{1}, V_{2}, V_{3})$ with respect to which $G$ is curious. In this section we will prove that given a hereditary class $X$ , if the bipartite graphs in $X$ are well-quasi-ordered by the labelled induced subgraph relation or have bounded clique-width, then the same is true for the curious graphs in $X$ .

A linear order $(x_{1}, x_{2}, \dots, x_{k})$ of the vertices of an independent set $I$ is

increasing if $i < j$ implies $N (x_{i}) \subseteq N (x_{j})$ ,
decreasing if $i < j$ implies $N (x_{i}) \supseteq N (x_{j})$ ,
monotone if it is either increasing or decreasing.

Bipartite graphs that are $2 P_{2}$ -free are also known as bipartite chain graphs. It is and it is well known (and easy to verify) that a bipartite graph $G$ is $2 P_{2}$ -free if and only if the vertices in each independent set of the bipartition admit a monotone ordering. Suppose $G$ is a curious graph with respect to some partition $(V_{1}, V_{2}, V_{3})$ . We say that (with respect to this partition) the graph $G$ is a curious graph of type $t$ if exactly $t$ of the graphs $G [V_{1} \cup V_{2}]$ , $G [V_{1} \cup V_{3}]$ and $G [V_{2} \cup V_{3}]$ contain an induced $2 P_{2}$ .

3.2.1 Curious Graphs of Type $0$ and $1$ .

Note that if $G$ is a curious graph of type $0$ or $1$ with respect to the partition $(V_{1}, V_{2}, V_{3})$ then without loss of generality, we may assume that $G [V_{1} \cup V_{2}]$ and $G [V_{1} \cup V_{3}]$ are both $2 P_{2}$ -free.

Lemma 12

Let $G$ be a curious graph with respect to $(V_{1}, V_{2}, V_{3})$ , such that $G [V_{1} \cup V_{2}]$ and $G [V_{1} \cup V_{3}]$ are both $2 P_{2}$ -free. Then the vertices of $V_{1}$ admit a linear ordering which is decreasing in $G [V_{1} \cup V_{2}]$ and increasing in $G [V_{1} \cup V_{3}]$ .

Proof

For a set $S \subseteq V$ , we use $N_{S} (u) := N (u) \cap S$ to denote the set of vertices in $S$ that are adjacent to $u$ . We may choose a linear order $x_{1}, \dots, x_{ℓ}$ of the vertices of $V_{1}$ according to their neighbourhood in $V_{2}$ , breaking ties according to their neighbourhood in $V_{3}$ i.e. an order such that:

if $N_{V_{2}} (x_{i}) ⊋ N_{V_{2}} (x_{j})$ then $i < j$ and
if $N_{V_{2}} (x_{i}) = N_{V_{2}} (x_{j})$ and $N_{V_{3}} (x_{i}) ⊊ N_{V_{3}} (x_{j})$ then $i < j$ .

Clearly such an ordering is decreasing in $G [V_{1} \cup V_{2}]$ .

Suppose, for contradiction, that this order is not increasing in $G [V_{1} \cup V_{3}]$ . Then there must be indices $i < j$ such that $N_{V_{3}} (x_{i}) ⊋ N_{V_{3}} (x_{j})$ . Then $N_{V_{2}} (x_{i}) \neq N_{V_{2}} (x_{j})$ by Property (ii). By Property (i) it follows that $N_{V_{2}} (x_{i}) ⊋ N_{V_{2}} (x_{j})$ . This means that there are vertices $y \in N_{V_{2}} (x_{i}) ∖ N_{V_{2}} (x_{j})$ and $z \in N_{V_{3}} (x_{i}) ∖ N_{V_{3}} (x_{j})$ . Now if $y$ is adjacent to $z$ then $G [x_{i}, y, z]$ is a rainbow $K_{3}$ and if $y$ is non-adjacent to $z$ then $G [x_{j}, y, z]$ is a rainbow $3 P_{1}$ . This contradiction implies that the order is indeed decreasing in $G [V_{1} \cup V_{3}]$ , which completes the proof.∎

Lemma 13

If $G$ is a curious graph of type $0$ or $1$ with respect to a partition $(V_{1}, V_{2}, V_{3})$ then $G$ can be partitioned into slices that are bipartite.

Proof

Let $x_{1}, \dots, x_{ℓ}$ be a linear order on $V_{1}$ satisfying Lemma 12. Let $V_{1}^{0} = \emptyset$ and for $i \in {1, \dots, ℓ}$ , let $V_{1}^{i} = {x_{i}}$ . We partition $V_{2}$ and $V_{3}$ as follows. For $i \in {0, \dots, ℓ}$ , let $V_{2}^{i} = {y \in V_{2} | x_{j} y \in E (G) if and only if j \leq i}$ . For $i \in {0, \dots, ℓ}$ , let $V_{3}^{i} = {z \in V_{3} | x_{j} z \notin E (G) if and only if j \leq i}$ . In particular, note that the vertices of $V_{2}^{ℓ} \cup V_{3}^{0}$ and $V_{2}^{0} \cup V_{3}^{ℓ}$ are complete and anti-complete to $V_{1}$ , respectively. The following properties hold:

If $j < k$ then $V_{1}^{j}$ is complete to $V_{2}^{k}$ and anti-complete to $V_{3}^{k}$ .
If $j > k$ then $V_{1}^{j}$ is anti-complete to $V_{2}^{k}$ and complete to $V_{3}^{k}$ .

If $j < k$ and $y \in V_{2}^{j}$ is non-adjacent to $z \in V_{3}^{k}$ then $G [x_{k}, y, z]$ is a rainbow $3 P_{1}$ , a contradiction. If $j > k$ and $y \in V_{2}^{j}$ is adjacent to $z \in V_{3}^{k}$ then $G [x_{j}, y, z]$ is a rainbow $K_{3}$ , a contradiction. It follows that:

If $j < k$ then $V_{2}^{j}$ is complete to $V_{3}^{k}$ .
If $j > k$ then $V_{2}^{j}$ is anti-complete to $V_{3}^{k}$ .

For $i \in {0, \dots, ℓ}$ , let $G^{i} = G [V_{1}^{i} \cup V_{2}^{i} \cup V_{3}^{i}]$ . The above properties about the edges between the sets $V_{j}^{i}$ show that $G$ can be partitioned into the slices $G^{0}, \dots, G^{ℓ}$ . Now, for each $i \in {0, \dots, ℓ}$ , $V_{1}^{i}$ is anti-complete to $V_{3}^{i}$ , so every slice $G^{i}$ is bipartite. This completes the proof.∎

3.2.2 Curious Graphs of Type $2$ and $3$ .

Lemma 14

Fix $t \in {2, 3}$ . If $G$ is a curious graph of type $t$ with respect to a partition $(V_{1}, V_{2}, V_{3})$ then $G$ can be partitioned into slices of type at most $t - 1$ .

Proof

Fix $t \in {2, 3}$ and let $G$ be a curious graph of type $t$ with respect to a partition $(V_{1}, V_{2}, V_{3})$ . We may assume that $G [V_{1} \cup V_{2}]$ contains an induced $2 P_{2}$ .

Claim 1. Given a $2 P_{2}$ in $G [V_{1} \cup V_{2}]$ , every vertex of $V_{3}$ has exactly two neighbours in the $2 P_{2}$ and these neighbours either both lie in $V_{1}$ or both lie in $V_{2}$ .
Let $x_{1}, x_{2} \in V_{1}$ and $y_{1}, y_{2} \in V_{2}$ induce a $2 P_{2}$ in $G$ such that $x_{1}$ is adjacent to $y_{1}$ but not to $y_{2}$ and $x_{2}$ is adjacent to $y_{2}$ but not to $y_{1}$ . Consider $z \in V_{3}$ . For $i \in {1, 2}$ , if $z$ is adjacent to both $x_{i}$ and $y_{i}$ , then $G [x_{i}, y_{i}, z]$ is a rainbow $K_{3}$ , a contradiction. Therefore $z$ can have at most one neighbour in ${x_{1}, y_{1}}$ and at most one neighbour in ${x_{2}, y_{2}}$ . If $z$ is non-adjacent to $x_{1}$ and $y_{2}$ then $G [x_{1}, y_{2}, z]$ is a rainbow $3 P_{1}$ , a contradiction. Therefore, $z$ can have at most one non-neighbour in ${x_{1}, y_{2}}$ , and similarly $z$ have have at most one non-neighbour in ${x_{2}, y_{1}}$ . Therefore, if $z$ is adjacent to $x_{1}$ , then it must be non-adjacent to $y_{1}$ , so it must be adjacent to $x_{2}$ , so it must be non-adjacent to $y_{2}$ . Similarly, if $z$ is non-adjacent to $x_{1}$ then it must be adjacent to $y_{2}$ , so it must be non-adjacent to $x_{2}$ , so it must be adjacent to $y_{1}$ . Hence Claim 3.2.2 follows.

Consider a maximal set ${H^{1}, \dots, H^{q}}$ of vertex-disjoint sets that induce copies of $2 P_{2}$ in $G [V_{1} \cup V_{2}]$ . We say that a vertex of $V_{3}$ distinguishes two graphs $G [H^{i}]$ and $G [H^{j}]$ if its neighbours in $H^{i}$ and $H^{j}$ do not belong to the same set $V_{k}$ . We group these sets $H^{i}$ into blocks $B^{1}, \dots, B^{p}$ that are not distinguished by any vertex of $V_{3}$ . In other words, every $B^{i}$ contains at least one $2 P_{2}$ and every vertex of $V_{3}$ is complete to one of the sets $B^{i} \cap V_{1}$ and $B^{i} \cap V_{2}$ and anti-complete to the other. For $j \in {1, 2}$ , let $B_{j}^{i} = B^{i} \cap V_{j}$ . We define a relation $<_{B}$ on the blocks as follows:

$B^{i} <_{B} B^{j}$ holds if $B_{1}^{i}$ is complete to $B_{2}^{j}$ , while $B_{2}^{i}$ is anti-complete to $B_{1}^{j}$ .

For distinct blocks $B^{i}$ , $B^{j}$ at most one of $B^{i} <_{B} B^{j}$ and $B^{j} <_{B} B^{i}$ can hold.

Claim 2. Let $B^{i}$ and $B^{j}$ be distinct blocks. There is a vertex $z \in V_{3}$ that differentiates $B^{i}$ and $B^{j}$ . If $z$ is complete to $B_{2}^{i} \cup B_{1}^{j}$ and anti-complete to $B_{1}^{i} \cup B_{2}^{j}$ then $B^{i} <_{B} B^{j}$ (see also Figure 3). If $z$ is complete to $B_{2}^{j} \cup B_{1}^{i}$ and anti-complete to $B_{1}^{j} \cup B_{2}^{i}$ then $B^{j} <_{B} B^{i}$ .
By definition of the blocks $B^{i}$ and $B^{j}$ there must be a vertex $z \in V_{3}$ that distinguishes them. Without loss of generality we may assume that $z$ is complete to $B_{2}^{i} \cup B_{1}^{j}$ and anti-complete to $B_{1}^{i} \cup B_{2}^{j}$ . It remains to show that $B^{i} <_{B} B^{j}$ . If $y \in B_{2}^{i}$ is adjacent to $z \in B_{1}^{j}$ then $G [x, y, z]$ is a rainbow $K_{3}$ , a contradiction. If $y \in B_{1}^{i}$ is non-adjacent to $z \in B_{2}^{j}$ then $G [x, y, z]$ is a rainbow $3 P_{1}$ , a contradiction. Therefore $B_{2}^{i}$ is anti-complete to $B_{1}^{j}$ and $B_{1}^{i}$ is complete to $B_{2}^{j}$ . It follows that $B^{i} <_{B} B^{j}$ . Claim 3.2.2 follows by symmetry.

Figure 3: Two blocks

B^{i}

and

B^{j}

with

B^{i} <_{B} B^{j}

and a vertex

z \in V_{3}

differentiating them.

Claim 3. The relation $<_{B}$ is transitive.
Suppose that $B^{i} <_{B} B^{j}$ and $B^{j} <_{B} B^{k}$ . Since $B^{j}$ and $B^{k}$ are distinct, there must be a vertex $z \in V_{3}$ that distinguishes them and combining Claim 3.2.2 with the fact that $B^{j} <_{B} B^{k}$ it follows that $z$ must be complete to $B_{2}^{j} \cup B_{1}^{k}$ . Suppose $x \in B_{1}^{i}$ and $y \in B_{2}^{j}$ . Then $x$ is adjacent to $y$ , since $B^{i} <_{B} B^{j}$ and $y$ is adjacent to $z$ by choice of $z$ . Now $x$ is must be non-adjacent to $z$ otherwise $G [x, y, z]$ would be a rainbow $K_{3}$ . Therefore $z$ is anti-complete to $B_{1}^{i}$ , so $z$ distinguishes $B^{i}$ and $B^{k}$ . By Claim 3.2.2 it follows that $B^{i} <_{B} B^{k}$ . This completes the proof of Claim 3.2.2.

Combining Claims 3.2.2–3.2.2, we find that $<_{B}$ is a linear order on the blocks. We obtain the following conclusion, which we call the chain property.

Claim 4. The set of blocks admits a linear order $B^{1} <_{B} B^{2} <_{B} \dots <_{B} B^{p}$ such that

if $i < j$ then $B_{1}^{i}$ is complete to $B_{2}^{j}$ , while $B_{2}^{i}$ is anti-complete to $B_{1}^{j}$ and
for each $z \in V_{3}$ there exists an $i \in {0, \dots, p}$ such that if $j \leq i$ then $z$ is complete to $B_{2}^{j}$ and anti-complete to $B_{1}^{j}$ and if $j > i$ then $z$ is anti-complete to $B_{2}^{j}$ and complete to $B_{1}^{j}$ .

Next consider the set of vertices in $V_{1} \cup V_{2}$ that do not belong to any set $B^{i}$ . Let $R$ denote this set and note that $G [R]$ is $2 P_{2}$ -free by maximality of the set ${H^{1}, \dots, H^{q}}$ . For $i \in {1, 2}$ let $R_{i} = R \cap V$

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1 Introduction

Conjecture 1 ([24])

Theorem 1.1

Theorem 1.2

Theorem 1.3 ([11])

2 Preliminaries

Lemma 1 ([8])

2.1 Clique-width

Lemma 2 ([7])

2.2 Well-quasi-orderability

Lemma 3 (Higman’s Lemma [17])

Lemma 4

Lemma 5 ([2])

Lemma 6 ([2, 20])

Lemma 7

Proof

2.3 k-uniform Graphs

Lemma 8 ([21])

Lemma 9

3 Partitioning 3-Partite Graphs

3.1 The Decomposition

Lemma 10

Proof

Lemma 11

Proof

3.2 Curious Graphs

3.2.1 Curious Graphs of Type 0 and 1.

Lemma 12

Proof

Lemma 13

Proof

3.2.2 Curious Graphs of Type 2 and 3.

Lemma 14

Proof

2.3 $k$ -uniform Graphs

3 Partitioning $3$ -Partite Graphs

3.2.1 Curious Graphs of Type $0$ and $1$ .

3.2.2 Curious Graphs of Type $2$ and $3$ .