Half-space separation in monophonic convexity

Abstract

We study half-space separation in the convexity of chordless paths of a graph, i.e., monophonic convexity. In this problem, one is given a graph and two (disjoint) subsets of vertices and asks whether these two sets can be separated by complementary convex sets, called half-spaces. While it is known this problem is $\mathbf{NP}$-complete for geodesic convexity -- the convexity of shortest paths -- we show that it can be solved in polynomial time for monophonic convexity.

Figures (11)

• Figure 1: A graph $G$ with two disjoint convex sets $A= \{a_1, a_2\}$ and $B = \{b_1, b_2\}$ (circled in green and blue resp.). $A$ and $B$ are not linked, but they can be linked along the path $a_1, v_1, v_3, b_1$ (in bold green / bold blue). Namely, $A \cup v_1$ and $B \cup v_3$ are linked and convex. Two half-spaces $H, \overline{H}$ separating $A \cup v_1$ and $B \cup v_3$ (hence $A$ and $B$) are drawn.
• Figure 2: A graph $G$ in which we seek to separate $A$ and $B$ (in circled green/blue resp.). The vertex $a_1$ is on a chordless $u_3b$-path (bold blue), so that $u_3 \in A / B$. Dually, $b$ is on a chordless $v_2a_2$-path (bold green), i.e., $v_2 \in B / A$. Besides, $h(v_1v_3)$ intersects both $A$ and $B$ (bold red). Thus, $v_1, v_3$ must be separated to separate $A$ and $B$, and $v_1v_3 \in \mathop{\mathrm{MFS}}\nolimits(A, B)$ holds.
• Figure 3: Pre-saturation applied to the sets $A$ and $B$ of Figure \ref{['fig:pre-saturation-1']}. For $\sigma(B, A)$, $u_1, u_2 \in B / A$ are added. For $\sigma(A, B)$, we have $u_3, u_4 \in A / B$ and $v_4 \in h(A \cup v_1) \cap h(A \cup v_3)$ (paths in bold green) with $v_1v_3 \in \mathop{\mathrm{MFS}}\nolimits(A, B)$.
• Figure 4: An example where pre-saturation can be applied twice. For $\sigma(A, B)$, $v_2$ is obtained from the forbidden pair $v_1v_3$. Once $v_2$ is added, $v_4, v_5$ become part of $\sigma(A, B) / \sigma(B, A)$. Observe that $B = \sigma(B, A) = \sigma(\sigma(B, A), \sigma(A, B))$. The remaining vertices $v_1, v_3$ can be separated in any way.
• Figure 5: Two cases where $A$ and $B$ are linked, convex and saturated. On the left (follow-up of Figure \ref{['fig:pre-saturation-2']}), $A$ and $B$ can be separated (two half-spaces are drawn). On the right, any bipartition of the vertices will contain one of the forbidden pair $v_1v_2, v_1v_3$ or $v_2v_3$. Thus, $A$ and $B$ are not separable.

Theorems & Definitions (48)

• Theorem 1: restate=THMseppoly, label=thm:separability-polynomial
• Remark 1
• Remark 2
• Theorem 2: dourado2010complexity, Theorem 2.1
• Lemma 1: gonzalez2020covering, Lemma 14
• Theorem 3: duchet1988convex, Theorem 5.1
• Theorem 4: dourado2010complexity, Theorem 4.1
• Proposition 1: restate=PROPabpath, label=prop:ab-path
• proof
• Lemma 2: restate=LEMlinked, label=lem:linked