On the Complexity of Finding Large Odd Induced Subgraphs and Odd Colorings

Abstract

We study the complexity of the problems of finding, given a graph G, a largest induced subgraph of G with all degrees odd (called an odd subgraph), and the smallest number of odd subgraphs that partition V(G). We call these parameters $\mathsf{mos}(G)$ and $\chi _{\mathsf{odd}}(G)$, respectively. We prove that deciding whether $\chi _{\mathsf{odd}}(G) \le q$ is polynomial-time solvable if $q \le 2$, and NP-complete otherwise. We provide algorithms in time $2^{{{\mathcal {O}}}(\mathsf{rw})} \cdot n^{{{\mathcal {O}}}(1)}$ and $2^{{{\mathcal {O}}}(q \cdot \mathsf{rw})} \cdot n^{{{\mathcal {O}}}(1)}$ to compute $\mathsf{mos}(G)$ and to decide whether $\chi _{\mathsf{odd}}(G) \le q$ on n-vertex graphs of rank-width at most $\mathsf{rw}$, respectively, and we prove that the dependency on rank-width is asymptotically optimal under the ETH. Finally, we give some tight bounds for these parameters on restricted graph classes or in relation to other parameters.

A Parity Version of Domination Problems

In this section we consider the “parity version” of domination problems. For simplicity, we just deal with the “odd” versions. Namely, an odd dominating set (resp. odd total dominating set) of a graph G is a set $S \subseteq V(G)$ such that every vertex in $V(G) \setminus S$ (resp. V(G)) has an odd number of neighbors in S. Accordingly, in the Minimum Odd Dominating Set (resp. Minimum Odd Total Dominating Set) problem, we are given a graph G and the objective is to find an odd dominating set (resp. odd total dominating set) in G of minimum size.

It is worth mentioning that what is usually called an “odd dominating set” in the literature (cf. for instance [9] and the references given in [10, 25]) differs from the definition given above. Indeed, in previous work a set $S \subseteq V$ is said to be an odd dominating set if $|N[v] \cap S|$ is odd for every vertex $v \in V(G)$. That is, vertices outside of S must have an odd number of neighbors in S, while vertices in S must have an even number of neighbors in S. Note that this definition differs from both definitions given in the above paragraph.

Concerning Odd Total Dominating Set, it was studied –among other parity problems– by Halldórsson et al. [26], who proved its NP-hardness by a reduction from the Codeword of Minimal Weight problem. However, the (quite involved) NP-hardness proof of this latter problem by Vardy [44] involves several nonlinear blow-ups, so a lower bound of $2^{o(n)}$ under the ETH cannot be deduced from it. Fortunately, we can indeed obtain a linear NP-hardness reduction for both Minimum Odd Dominating Set and Minimum Odd Total Dominating Set by doing simple local modifications to the proof of Sutner [43, Theorem 3.2] for a variant of odd total domination, which is from the 3-Sat problem. We omit the details here.

Once we know that none of these problems can be solved in time $2^{o(n)}$ on n-vertex graphs under the ETH, our main contribution in this section is to adapt the dynamic programming algorithms presented in Sect. 4 to solve both Minimum Odd Dominating Set and Minimum Odd Total Dominating Set in single-exponential time parameterized by the rank-width of the input graph. Namely, we prove the following two results.

Theorem 9

Given a graph G along with a decomposition tree of rank-width w, the Minimum Odd Dominating Set problem can be solved in time ${{\mathcal {O}}}^*(2^{3\mathsf{rw}})$.

Proof

The algorithm and its proof are nearly identical to the ones for Maximum Odd Subgraph (cf. Theorem 4), replacing condition $(\maltese )$ with

$$\begin{aligned}T_A[R,R'] = \underset{S\subseteq A}{\max }\{ |S| : S \equiv _2^A R \wedge \{v\in A\setminus S: |N(v)\cap S| \text { is even}\} = A\cap N_2(R')\setminus S\},\end{aligned}$$

and the leaves are instead defined as $T_{\{u\}}[\emptyset ,\{v\}] = \emptyset$ and $T_{\{u\}}[\{u\},\emptyset ]= T_{\{u\}}[\{u\},\{v\}] = \{u\}$. $T_{\{u\}}[\emptyset ,\emptyset ]$ is left empty. $\square$

Theorem 10

Given a graph G along with a decomposition tree of rank-width w, the Minimum Odd Total Dominating Set problem can be solved in time ${{\mathcal {O}}}^*(2^{3\mathsf{rw}})$.

Proof

Again, the algorithm and its proof are nearly identical to the ones for Maximum Odd Subgraph (cf. Theorem 4), replacing condition $(\maltese )$ with

$$\begin{aligned}T_A[R,R'] = \underset{S\subseteq A}{\max }\{ |S| : S \equiv _2^A R \wedge \{v\in A: |N(v)\cap S| \text { is even}\} = A\cap N_2(R')\},\end{aligned}$$

and the leaves are instead defined as $T_{\{u\}}[\emptyset ,\{v\}] = \emptyset$ and $T_{\{u\}}[\{u\},\{v\}] = \{u\}$. $T_{\{u\}}[\emptyset ,\emptyset ]$ and $T_{\{u\}}[\{u\},\emptyset ]$ are left empty. $\square$

As future work, it would be interesting to adapt the above algorithms to deal with the connected version of both problems, where the (total) odd dominating set S is further required to induce a connected graph; see [10] for related work about this variant of domination.