The Rank-Width of Edge-Coloured Graphs

Abstract

A C-coloured graph is a graph, that is possibly directed, where the edges are coloured with colours from the set C. Clique-width is a complexity measure for C-coloured graphs, for finite sets C. Rank-width is an equivalent complexity measure for undirected graphs and has good algorithmic and structural properties. It is in particular related to the vertex-minor relation. We discuss some possible extensions of the notion of rank-width to C-coloured graphs. There is not a unique natural notion of rank-width for C-coloured graphs. We define two notions of rank-width for them, both based on a coding of C-coloured graphs by ${\mathbb{F}}^{*}$-graphs—$\mathbb {F}$-coloured graphs where each edge has exactly one colour from $\mathbb{F}\setminus \{0\},\ \mathbb{F}$ a field—and named respectively $\mathbb{F}$ -rank-width and $\mathbb {F}$ -bi-rank-width. The two notions are equivalent to clique-width. We then present a notion of vertex-minor for $\mathbb{F}^{*}$-graphs and prove that $\mathbb{F}^{*}$-graphs of bounded $\mathbb{F}$-rank-width are characterised by a list of $\mathbb{F}^{*}$-graphs to exclude as vertex-minors (this list is finite if $\mathbb{F}$ is finite). An algorithm that decides in time O(n ³) whether an $\mathbb{F}^{*}$-graph with n vertices has $\mathbb{F}$-rank-width (resp. $\mathbb{F}$-bi-rank-width) at most k, for fixed k and fixed finite field $\mathbb{F}$, is also given. Graph operations to check MSOL-definable properties on $\mathbb{F}^{*}$-graphs of bounded $\mathbb{F}$-rank-width (resp. $\mathbb{F}$-bi-rank-width) are presented. A specialisation of all these notions to graphs without edge colours is presented, which shows that our results generalise the ones in undirected graphs.

Appendix: Proofs of Propositions 3.12 and 4.4

If $R\subseteq\{1,\ldots,k\}\times\{1,\ldots,k\}\times \mathbb{F}$, we let ◯ _R be the composition of the functions $add_{i,j}^{a}$ with (i,j,a)∈R. This notation is non ambiguous because $add_{i,j}^{a}\circ add_{k,l}^{b} =add_{k,l}^{b}\circ add_{i,j}^{a}$.

Proof of Proposition 3.12.

(1) Assume $G=\mathit{val}(t)$ for some term t in $T(\mathcal{F}_{k}^{\mathbb{F}},\mathcal{C}_{k})$. In order to prove that is a layout of V _G of $\operatorname {cutrk}^{{\mathbb{F}}}_{G}$-width at most k, it is enough to prove that for every subgraph H of G that is a value of a sub-term t′ of t, $\operatorname {cutrk}^{{\mathbb{F}}}_{G}(V_{H})\leq k$. But, by the definition of operations in $\mathcal{F}_{k}^{\mathbb{F}}$, the sub-matrix M _G [V _H ,V _G ∖V _H ] has at most k distinct rows. Thus, $\operatorname {cutrk}^{{\mathbb{F}}}_{G}(V_{H}) =\operatorname{rk}({ M_{G}}[{V_{H}},{V_{G}\setminus V_{H}}])\leq k$.

(2) Assume $(T,\mathcal{L})$ is a layout of V _G of $\operatorname {cutrk}^{{\mathbb{F}}}_{G}$-width k. Then, by Theorem 5.4 we can construct in time O(k ²⋅|V _G |²) a term t in $T(\mathcal{R}_{k}^{(\mathbb{F},\sigma)},\mathcal{C}_{k}^{\mathbb{F}})$ such that $G=\mathit{val}(t)$ and . We will construct inductively, on the size of t, a term t′ in $T(\mathcal{F}_{k'}^{\mathbb{F}},\mathcal{C}_{k'})$ with k′≤2⋅q ^k−1 such $G=\mathit{val}(t')$. We let $\beta:\mathbb{F}^{k}\to \{1,\ldots, q^{k}\}$ be a bijective function that enumerates the set of vectors in $\mathbb{F}^{k}$ with β(O _1,k )=1. We let {2′,…,(q ^k)′} be a disjoint copy of the set {2,…,q ^k}. If $t=\mathbf {u}$, then we let $t':=\mathbf {\beta(u)}$. Suppose then that t=t ₁⊗ _M,N,P t ₂. Then, we let

where R:={(β(u),β(v)′,λ)∣u⋅M⋅σ(v)^T=λ}, R′:={(β(v)′,β(u),σ(λ))∣u⋅M⋅σ(v)^T=λ}, h:{1,…,q ^k}→{1}∪{2′,…,(q ^k)′} is such that h(1)=1 and h(i):=i′, g:{1,…,q ^k}→{1,…,q ^k} is such that g(i):=β(β ⁻¹(i)⋅N), g′:{1}∪{2′,…,(q ^k)′}→{1,…,q ^k} is such that g′(1):=1 and g′(i′):=β(β ⁻¹(i)⋅P).

It is a straightforward induction to check that $G=\mathit{val}(t')$ and that . □

Proof of Proposition 4.4.

(1) Assume $G=\mathit{val}(t)$ for some term t in $T(\mathcal{F}_{k}^{\mathbb{F}},\mathcal{C}_{k})$. In order to prove that is a layout of V _G of $\operatorname {bicutrk}^{{\mathbb{F}}}_{G}$-width at most k, it is enough to prove that for every subgraph H of G that is a value of a sub-term t′ of t, $\operatorname {bicutrk}^{{\mathbb{F}}}_{G}(V_{H})\leq2k$. But, by the definition of operations in $\mathcal{F}_{k}^{\mathbb{F}}$, the sub-matrices M _G [V _H ,V _G ∖V _H ] and M _G [V _G ∖V _H ,V _H ] have at most k distinct rows. Thus, $\operatorname {bicutrk}^{{\mathbb{F}}}_{G}(V_{H}) =\operatorname{rk}({ M_{G}}[{V_{H}},{V_{G}\setminus V_{H}}]) +\operatorname{rk}({ M_{G}}[{V_{G}\setminus V_{H}},{V_{H}}]) \leq2k$.

(2) Assume $(T,\mathcal{L})$ is a layout of V _G of $\operatorname {bicutrk}^{{\mathbb{F}}}_{G}$-width k. Then, by Theorem 5.4 we can construct in time O(k ²⋅|V _G |²) a term t in $T(\mathcal{BR}_{k}^{\mathbb{F}},\mathcal{BC}_{k}^{\mathbb{F}})$ such that $G=\mathit{val}(t)$ and . We will construct inductively, on the size of t, a term t′ in $T(\mathcal{F}_{k'}^{\mathbb{F}},\mathcal{C}_{k'})$ with k′≤2⋅q ^k−1 such $G=\mathit{val}(t')$. For each pair (k ₁,k ₂) with k ₁+k ₂≤k, we let $\alpha_{k_{1},k_{2}}:\mathbb{F}^{k_{1}}\times \mathbb{F}^{k_{2}}\to\{1,\ldots,q^{k_{1}+k_{2}}\}$ be a bijective function that enumerates the set of pairs of vectors in $\mathbb{F}^{k_{1}}\times \mathbb{F}^{k_{2}}$ with $\alpha_{k_{1},k_{2}}((O_{1,k_{1}},O_{1,k_{2}}))=1$. We let {2′,…,(q ^k)′} be a disjoint copy of the set {2,…,q ^k}.

If $t=\mathbf {u\cdot v}$, then we let $t':=\mathbf {\alpha_{1,1}((u,v))}$. Suppose now that $t=t_{1}\otimes_{M_{1},M_{2},N_{1},N_{2},P_{1},P_{2}} t_{2}$ with M ₁,M ₂,N ₁,N ₂,P ₁,P ₂ being respectively k ₁×ℓ ₁, k ₂×ℓ ₂, $k_{1}\times k'_{1}$, $k_{2}\times k'_{2}$, $\ell_{1}\times k'_{1}$ and $\ell_{2}\times k'_{2}$-matrices. Then, we let

where $R:= \{(i,j',c)\mid i=\alpha_{k_{1},k_{2}}((u_{1},u_{2})),j=\alpha_{\ell_{1},\ell_{2}}((v_{1},v_{2}))\ \textrm{and}\ u_{1}\cdot M_{1}\cdot v_{1}^{T}=c\}$, $R':= \{(j',i,c)\mid j=\alpha_{\ell_{1},\ell_{2}}((v_{1},v_{2})),i=\alpha_{k_{1},k_{2}}((u_{1},u_{2})),\ \textrm{and}\ u_{2}\cdot M_{2}\cdot v_{2}^{T}=c\}$, h:{1,…,q ^k}→{1}∪{2′,…,(q ^k)′} is such that h(1)=1 and h(i):=i′, g:{1,…,q ^k}→{1,…,q ^k} is such that if $\alpha_{k_{1},k_{2}}^{-1}(i)=(u_{1},u_{2})$, then $g(i) :=\alpha_{k'_{1},k'_{2}}((u_{1}\cdot N_{1},u_{2}\cdot N_{2}))$, g′:{1}∪{2′,…,(q ^k)′}→{1,…,q ^k} is such that g′(1):=1 and if $\alpha_{\ell_{1},\ell_{2}}^{-1}(i) =(v_{1},v_{2})$, then $g'(i') := \alpha_{k'_{1},k'_{2}}((v_{1}\cdot P_{1},v_{2}\cdot P_{2}))$.

An easy induction shows that $G=\mathit{val}(t')$ and that . □