The Interplay Between Domination and Separation in Graphs

In the literature, several identification problems in graphs have been studied, of which, the most widely studied are the ones based on dominating sets as a tool of identification. Hereby, the objective is to separate any two vertices of a graph by their unique neighborhoods in a suitably chosen dominating or total-dominating set. Such a (total-)dominating set endowed with a separation property is often referred to as a code of the graph. In this paper, we study the four separation properties location, closed-separation, open-separation and full-separation. We address the complexity of finding minimum separating sets in a graph and study the interplay of these separation properties with several codes (establishing a particularly close relation between separation and codes based on domination) as well as the interplay of separation and complementation (showing that location and full-separation are the same on a graph and its complement, whereas closed-separation in a graph corresponds to open-separation in its complement).

💡 Research Summary

The paper investigates four separation properties in simple undirected graphs—Location (L), Closed‑separation (I), Open‑separation (O), and Full‑separation (F)—and studies their interaction with domination concepts. A dominating set (D‑set) requires every vertex to have a neighbor (or itself) in the set, while a total‑dominating set (TD‑set) requires every vertex to have a neighbor in the set. Combining each separation property with either domination or total‑domination yields eight well‑known identifying codes: LD, LTD, OD, OTD, ID, ITD, FD, and FTD.

The authors first prove that finding a minimum S‑set (a set satisfying one of the four separation properties) is NP‑complete for every S∈{L,O,I,F}. The reduction is from the classic Test‑Cover problem. For a given instance (U,T,ℓ) they construct a graph G_S by (1) creating a base vertex set Q consisting of r(S) copies of each item and a small auxiliary set R, (2) adding a vertex w(T) for each test and connecting it to the copies of items belonging to that test, (3) attaching an S‑gadget (a small graph with designated vertices B_S) to each w(T) and to a special gadget for the whole universe, and (4) setting the target size k = ℓ + p(S)·|T| + q(S). For I‑sets the construction uses r=1, a clique on Q, and a P₆ gadget; analogous adaptations work for L, O, and F. Lemma 2.1 shows that any I‑set must contain at least 4|T|+3 vertices from the gadgets, establishing the correspondence between a test cover of size ≤ℓ and an I‑set of size ≤k. Consequently, Min S‑Set is NP‑complete for all four separation notions.

Next, the paper relates the sizes of minimum S‑sets (γ_S) to the sizes of the eight codes (γ_X). Trivial inequalities such as γ_L ≤ γ_O and γ_I ≤ γ_F hold for any admissible graph. More substantial bounds are proved:

• γ_SD(G) ≤ γ_S(G)+1 for any S,
• γ_STD(G) ≤ γ_S(G)+1 when S∈{O,F},
• γ_STD(G) ≤ 2·γ_S(G) when S∈{L,I}. These results are illustrated on the family of thin headless spiders H_k (a clique Q of size k together with an independent set S of size k, each q_i adjacent only to s_i). For k≥4 the authors compute exact values: γ_L(H_k)=γ_O(H_k)=k−1, γ_I(H_k)=k+1, γ_F(H_k)=2k−2. These values match or differ by at most one from the corresponding code numbers known from earlier work, confirming the tightness of the bounds.

The paper then examines how separation numbers behave under graph complementation. The main theorems state: (a) γ_L(G)=γ_L(Ĝ) for any graph, (b) if G has no closed twins then γ_I(G)=γ_O(Ĝ), (c) if G has no open twins then γ_O(G)=γ_I(Ĝ), (d) if G has no twins at all then γ_F(G)=γ_F(Ĝ). From these, corollaries follow for the eight codes: the LD‑numbers of a graph and its complement differ by at most one, and similarly for the pairs (ID,OD), (OD,ID), and (FD,FTD) under the appropriate twin‑free conditions.

The authors conclude by emphasizing that minimum S‑sets provide a unifying framework linking domination‑based identifying codes and separation properties. Since all Min S‑Set problems are NP‑complete, exact algorithms are unlikely for general graphs, motivating the study of special graph classes, approximation schemes, and parameterized algorithms. Future directions suggested include deriving closed formulas for γ_S and γ_X on trees, planar graphs, and other restricted families; improving the constant factors in the bounds; exploring dynamic updates of codes under edge/vertex modifications; and deepening the understanding of twin structures in relation to complementarity.

Overall, the paper offers a comprehensive theoretical treatment of how domination and separation intertwine, establishes hardness results, provides tight bounds, and uncovers elegant complementarity relations, thereby extending the toolkit for graph identification problems.