Edge-switching homomorphisms of edge-coloured graphs

Brewster, Richard C.; Graves, Timothy

doi:10.1016/j.disc.2008.03.021

Cited by 26 publications

(37 citation statements)

References 6 publications

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“…Recently, the theory of s-homomorphisms was more extensively developed in [24] (see also [9,23,25] for subsequent studies). A related notion of homomorphisms of edge-coloured graphs where there is a switching operation is studied in [4,20,21]. See also [17] for homomorphisms of digraphs where a similar switching operation is allowed.…”

Section: S-homomorphismsmentioning

confidence: 99%

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The complexity of signed graph and edge-coloured graph homomorphisms

Brewster

Foucaud

Hell

et al. 2017

Discrete Mathematics

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We study homomorphism problems of signed graphs from a computational point of view. A signed graph (G, Σ) is a graph G where each edge is given a sign, positive or negative; Σ ⊆ E(G) denotes the set of negative edges. Thus, (G, Σ) is a 2-edge-coloured graph with the property that the edge-colours, {+, −}, form a group under multiplication. Central to the study of signed graphs is the operation of switching at a vertex, that results in changing the sign of each incident edge. We study two types of homomorphisms of a signed graph (G, Σ) to a signed graph (H, Π): ec-homomorphisms and s-homomorphisms. Each is a standard graph homomorphism of G to H with some additional constraint. In the former, edge-signs are preserved. In the latter, edge-signs are preserved after the switching operation has been applied to a subset of vertices of G.We prove a dichotomy theorem for s-homomorphism problems for a large class of (fixed) target signed graphs (H, Π). Specifically, as long as (H, Π) does not contain a negative (respectively a positive) loop, the problem is polynomial-time solvable if the core of (H, Π) has at most two edges, and is NP-complete otherwise. (Note that this covers all simple signed graphs.) The same dichotomy holds if (H, Π) has no negative digons, and we conjecture that it holds always. In our proofs, we reduce s-homomorphism problems to certain ec-homomorphism problems, for which we are able to show a dichotomy. In contrast, we prove that a dichotomy theorem for ec-homomorphism problems (even when restricted to bipartite target signed graphs) would settle the dichotomy conjecture of Feder and Vardi.

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Section: S-homomorphismsmentioning

confidence: 99%

“…We now describe a construction that is crucial in our proofs. [4] in a more general setting related to permutations (they used the term permutation graph). Their work built on that of Klostermeyer and MacGillivray [17] who used a similar definition in the context of digraphs.…”

Section: Switching Graphsmentioning

confidence: 99%

The complexity of signed graph and edge-coloured graph homomorphisms

Brewster

Foucaud

Hell

et al. 2017

Discrete Mathematics

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“…Let u be a vertex of G that is not in K 2,5 . Suppose u is in the face ax i bx i+1 of K 2,5 . Then x i+3 is not adjacent to u.…”

Section: Claimmentioning

confidence: 99%

Homomorphisms of Signed Graphs

Naserasr

Rollová

Sopena

2014

Journal of Graph Theory

146

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A signed graph [G, ] is a graph G together with an assignment of signs + and − to all the edges of G where is the set of negative edges. Furthermore [G, 1 ] and [G, 2 ] are considered to be equivalent if the symmetric difference of 1 and 2 is an edge cut of G. Naturally arising from matroid theory, several notions of graph theory, such as the theory of minors and the theory of nowhere-zero flows, have been already extended to signed graphs. In an unpublished manuscript, B. Guenin introduced the notion of signed graph homomorphisms where he showed how

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“…Proof Brewster and Graves , Theorem 12] obtained a general result on m ‐edge‐colored graphs, which gives the following in the case

m = 2

: a graph

(G, s)

admits a switching homomorphism to a graph

(H, t)

if and only if

A T (G, s)

admits a 2‐edge‐colored homomorphism to

A T (H, t)

. Notice that

A T (G, s)

admits a 2‐edge‐colored homomorphism to

A T (H, t)

if and only if

(G, s)

admits a 2‐edge‐colored homomorphism to

A T (H, t)

.…”

Section: Target Graphsmentioning

confidence: 99%

“…Proof. Brewster and Graves [4], Theorem 12] obtained a general result on m-edgecolored graphs, which gives the following in the case m = 2: a graph (G, s) admits a switching homomorphism to a graph (H, t ) if and only if AT (G, s) admits a 2-edgecolored homomorphism to AT (H, t ).…”

Section: Lemma 22 a Graph (G S) Admits A Switching Homomorphism Tomentioning

confidence: 99%